Surfaces energies

Surface tension = y Change in area = dA = Idx Change in x-direction = dx 

One may consider another example to describe surface energy. Let us imagine that a liquid fills a container of the shape of a funnel. In the funnel, if one moves the liquid upwards, then there will be an increase in surface area. This requires that some molecules from the bulk phase have to move into the surface area and create extra surface The work required to do so will be (force x area) y As. This work is reversible at constant temperature and pressure, and thus gives the increase in free energy of the system  

the tension per unit length in a single surface, or surface tension y, is numerically equal to the surface energy per unit area. Then Gs, the surface free energy per unit area is 

Under reversible conditions, the heat (q) associated with it gives the surface entropy Ss  

The quantity Es has been found to provide more useful information on surface phenomena than any of the other quantities. 

The surface energy 7 of a solid is the amount of energy needed to create a unit area of new surface. The process can be pictured as shown in Fig. 4.7a. where two new surfaces are created by cutting a solid in two. Given this simple picture, the surface energy is simply the product of the number of bonds broken per unit area of crystal surface and the energy per bond 

For the sake of simplicity, only first-neighbor interactions will be considered here, which implies that bond is given by Eq. (2.15). Also note that since. V, is a function of crystallography, it follows that 7 is also a function of crystallography. 

To show how to calculate surface energies by starting with Eq. (4.15). 

Substance Surface Environment Temp.. K Surface energy. J/m  

The minus sign is introduced because energy has to be consumed to create a surface. Calculations of surface energies based on Eq. (4.16) invariably yield values that are substantially greater than the measured ones (see Table 4.4). The reason for this discrepancy comes about because in the simple model, surface relaxation and rearrangement of the atoms upon the formation of the new surface were not allowed. When the surface is allowed to relax, much of the energy needed to form it is recovered, and the theoretical predictions do indeed approach the experimentally measured ones. 

FIGURE 2.19 The concept of surface energy, (a) A molecule immersed in the bulk of a condensed phase is subject to attractive interactions in all directions (depicted as arrows), (b) When a molecule is brought to the surface in the presence of a dilute gas or vacuum, attractive forces in the direction normal to the surface (dotted arrow) are missing, so that a net work should be done. 

Note that, for a thermodynamically stable interface, this should be a positive quantity otherwise, the surface area will spontaneously increase without stopping until the two phases get mixed. If the second phase is a diluted gas such as air, the interfacial energy will be close to that in the presence of vacuum, so that customarily one speaks in this case of surface energy (or surface tension). 

The extensive variable of the surface energy is the area and the intensive variable is the surface tension. The surface energy is a typical example for an energy form, where the extensive variable can be increased and decreased nearly arbitrarily. 

We first deal with a liquid in contact with vacuum. The surface tension a is a vector that points into the inner region of the liquid. In static equilibrium, the vector points normal to the surface, whereas the surface vector dA which is also a normal vector points outward from the liquid. Since the two vectors are opposite in direction, the negative sign in the energy equation cancels. 

If the liquid is not in contact with vacuum, but with a vapor or a liquid, then the above considerations hold. However, we must see the situation from both phases. The surface tension will be modified by the neighborhood of another phase, even in a different way for the individual phases. The particular change of the surface tension also governs the miscibility. Since both phases are coupled in the case by a 

A knowledge ofthe surface tension of the liquid adhesive and the surface energy of the substrate to which it will be applied is important in predicting the wettability and subsequent adhesion ofthe cured adhesive. As a rule, the surface tension ofthe adhesive should be less than the surface energy ofthe substrate. Additives are often added to adhesive formulations to reduce their surface tensions, while cleaning processes are used to increase surface energies of substrates. 

In order to introduce new surfaces into an object, a certain amount of energy must first be expended. In this regard, it can be said that surfaces - whatever they are - are 

1) This choice of the system is made to simplify the discussion, because in this way each atom represented by a sphere from the hard sphere model corresponds directly to the crystallographic lattice points. In principle, however, the surface energy of any system can be calculated with the same scheme. 

Once the number of broken bonds per unit cell of a surface has been determined, the surface energy can be estimated from the following equation  

For this matter, Mackenzie et al. [14] developed a systematic way to calculate surface energies based on the broken bond model. The basic idea is that, since all the bonds to be broken for opening a surface should lie in the same direction of the surface normal, the total number of such bonding vectors, normalized by the area of the unit cell of the surface, is equal to the product, NS-NB. For pure metals with the fee symmetry, the energy of the (hkl) surface is given as  

Although the discussion on the broken bond model has been mainly focused on pure metallic surfaces, the extension can be easily made simply by replacing the binding energy term Ej, in Eq. (1) with the interaction energies among different atoms in the case of metallic alloys [15-17]. However, in the case of ceramic materials the 

Capillary forces were used to pump reagents through Si microchannels [397,459]. Additional gradients in surface pressure, which could be created by electrochem-ically generating and consuming surface-active species at the two ends of a channel, have been used for liquid pumping [369]. 

Surface energy present in a small liquid drop at the inlet reservoir was used to pump the liquid through a PDMS microchannel [398]. 

The surface energies of (100), (111), and (110) of diamond in a plasma environment (in the presence of H atoms and at high temperature) were evaluated using simple assumptions and an equation [98]. In the surface energy versus 7 diagram for 

The quantity y represents the force per unit length of the surface (mN/m = dyn/cm), and this force is defined as surface tension or IFT. Surface tension, y, is the differential change of free energy with the change of surface area at constant temperature, pressure, and composition. 

FIGURE 1.11 Surface film of a liquid (schematic) (see text for details). 

In what follows surface is defined as the plane between condensed matter and a vapour phase or vacuum, such as solid/vapour and liquid/vapour interfaces. In a broader sense, the term interface is used for the dividing plane between any two different phases. The existence of an interface means, by itself, the presence of an excess interface energy over the bulk energy. Since the driving force for sintering is the reduction of the total interfacial energy of the system concerned, it will be useful to understand the thermodynamic characteristics of interfacial energy. 

may be regarded as an additional term due to the presence of a transition layer which is determined by the curvature and area of the interface. 

an infinitesimal change in the internal energy at the surface, which is an excess surface quantity, can be expressed as 

Equation (2.4), the thermodynamic definition of y, shows that y is the reversible work required to create a unit area of the surface. 

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Ramsay-Shields equation An equation relating the molecular surface energy of a liquid with its temperature... 

Often, and as a good approximation, and the surface energy are not distinguished, so Eq. III-6 can be seen in the form... 

The total surface energy generally is larger than the surface free energy. It is frequently the more informative of the two quantities, or at least it is more easily related to molecular models. 

Fig. III-2. Variation of surface tension and total surface energy of CCU with temperature. (Data from Ref. 2.)...

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 surface free energy can be regarded as the work of bringing a molecule from the interior of a liquid to the surface, and that this work arises from the fact that, although a molecule experiences no net forces while in the interior of the bulk phase, these forces become unbalanced as it moves toward the surface. As discussed in connection with Eq. Ill-IS and also in the next sections, a knowledge of the potential function for the interaction between molecules allows a calculation of the total surface energy if this can be written as a function of temperature, the surface free energy is also calculable. 

The interface between a solid and its vapor (or an inert gas) is discussed in this chapter from an essentially phenomenological point of view. We are interested in surface energies and free energies and in how they may be measured or estimated theoretically. The study of solid surfaces at the molecular level, through the methods of spectroscopy and diffraction, is taken up in Chapter VIII. 

The immobility of the surface atoms of a refractory solid has the consequence that the surface energy and other physical properties depend greatly on the immediate history of the material. A clean cleavage surface of a crystal will have a different (and probably lower) surface energy than a ground, abraded, heat-treated or polished surface of the same material. 

Theoretical Estimates of Surface Energies and Free Energies... 

Face-centered cubic crystals of rare gases are a useful model system due to the simplicity of their interactions. Lattice sites are occupied by atoms interacting via a simple van der Waals potential with no orientation effects. The principal problem is to calculate the net energy of interaction across a plane, such as the one indicated by the dotted line in Fig. VII-4. In other words, as was the case with diamond, the surface energy at 0 K is essentially the excess potential energy of the molecules near the surface. 

The calculation is made by determining the primary contribution to the surface energy, that of the two separate parts, holding all the atoms in fixed positions. The total energy is reduced by the rearrangement of the surface layer to its equilibrium position as... 

The uncertainties in choice of potential function and in how to approximate the surface distortion contribution combine to make the calculated surface energies of ionic crystals rather uncertain. Some results are given in Table VII-2, but comparison between the various references cited will yield major discrepancies. Experimental verification is difficult (see Section VII-5). Qualitatively, one expects the surface energy of a solid to be distinctly higher than the surface tension of the liquid and, for example, the value of 212 ergs/cm for (100)... 

Calculated Surface Energies and Surface Stresses (ergs/cm )... 

The calculation of the surface energy of metals has been along two rather different lines. The first has been that of Skapski, outlined in Section III-IB. In its simplest form, the procedure involves simply prorating the surface energy to the energy of vaporization on the basis of the ratio of the number of nearest neighbors for a surface atom to that for an interior atom. The effect is to bypass the theoretical question of the exact calculation of the cohesional forces of a metal and, of course, to ignore the matter of surface distortion. 

The second model is a quantum mechanical one where free electrons are contained in a box whose sides correspond to the surfaces of the metal. The wave functions for the standing waves inside the box yield permissible states essentially independent of the lattice type. The kinetic energy corresponding to the rejected states leads to the surface energy in fair agreement with experimental estimates [86, 87],... 

Factors Affecting the Surface Energies and Surface Tensions of Actual Crystals... 

It might be noted that only for particles smaller than about 1 /ig or of surface area greater than a few square meters per gram does the surface energy become significant. Only for very small particles does the edge energy become important, at least with the assumption of perfect cubes. 

Side (cm) Total Area (cm ) Total Edge (cm) Surface Energy (ergs/g) Edge Energy (ergs/g)... 

It is because of these complications, both theoretical and practical, that it is doubtful that calculated surface energies for solids will ever serve as more than a guide as to what to expect experimentally. Corollaries are that different preparations of the same substance may give different values and that widely different experimental methods may yield different apparent values for a given preparation. In this last connection, see Section VII-5 especially. 

Gilman [124] and Westwood and Hitch [135] have applied the cleavage technique to a variety of crystals. The salts studied (with cleavage plane and best surface tension value in parentheses) were LiF (100, 340), MgO (100, 1200), CaFa (111, 450), BaFj (111, 280), CaCOa (001, 230), Si (111, 1240), Zn (0001, 105), Fe (3% Si) (100, about 1360), and NaCl (100, 110). Both authors note that their values are in much better agreement with a very simple estimate of surface energy by Bom and Stem in 1919, which used only Coulomb terms and a hard-sphere repulsion. In more recent work, however, Becher and Freiman [126] have reported distinctly higher values of y, the critical fracture energy. ... 

Westwood and Hitch suggest, incidentally, that the cleavage experiment, not being fully reversible, may give only a bond-breaking or nearest-neighbor type of surface energy with little contribution from surface distortion. 

The illustrative data presented in Table VII-3 indicate that the total surface energy may amount to a few tenths of a calorie per gram for particles on the order of 1 /xm in size. When the solid interface is destroyed, as by dissolving, the surface energy appears as an extra heat of solution, and with accurate calorimetry it is possible to measure the small difference between the heat of solution of coarse and of finely crystalline material. 

D. Dependence of Other Physical Properties on Surface Energy Changes at a Solid Interface... 

Make the following approximate calculations for the surface energy per square centimeter of solid krypton (nearest-neighbor distance 3.97 A), and compare your results with those of Table VII-1. (a) Make the calculations for (100), (110), and (111) planes, considering only nearest-neighbor interactions, (b) Make the calculation for (100) planes, considering all interactions within a radius defined by the sum... 

Calculate the surface energy at 0 K of (100) planes of radon, given that its energy of vaporization is 35 x 10 erg/atom and that the crystal radius of the radon atom is 2.5 A. The crystal structure may be taken to be the same as for other rare gases. You may draw on the results of calculations for other rare gases. 

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