Liquid drop theory

During recent decades a great amount of knowledge about the properties of atomic nuclei has been gathered. An extensive theory of nucleonic interactions and nuclear structure [liquid-drop theory (7), shell theory (2, 3), unified theory (4), cluster theory (5—7)] has been developed... 

Figure 9.5 Molecular dynamics calculations of the nondimcnsional surface pressure (difference between pressure inside a drop and the gas) for a Lennard-Jones intermolecular potential. Classical liquid drop theory begins to break down fordroplei radii smallcrihan about 10 times the Lennard-Jones diameter CTij. Calculations for Ar/su = 0.71 and = 0,58. (After Thompson et al, 1984.)...

Weakleim, C. L., and Reiss, H. (1993) Toward a molecular theory of vapor-phase nucleation. III. Thermodynamic properties of argon clusters from Monte Carlo simulations and a modified liquid drop theory, J. Chem. Phys. 99, 5374-5383. 

The earliest theory of nuclear structure was the liquid drop theory, developed around 1940. It pictured the nucleus as a structureless drop of liquid in which repulsions between positive charges are opposed by a kind... 

The work of Reiss and co-workers puts the question of the equilibrium distribution of liquid embryos in dilute supercooled vapors on sound conceptual ground. However, having to calculate embryo free energies by simulation rules out the use of such an approach in practical applications. To overcome this limitation, Weakliem and Reiss [67] developed a modified liquid drop theory that combines elements of the physically consistent cluster with the conventional capillarity approximation. These same authors have also developed a rate theory which allows the calculation of nucleation rates in supercooled vapors [68]. The dependence of the predicted rates on supersaturation agree with classical nucleation theory, but the temperature dependence shows systematic deviations, in accordance with scaling arguments [54]. 

Liquids are able to flow. Complicated stream patterns arise, dependent on geometric shape of the surrounding of the liquid and of the initial conditions. Physicists tend to simplify things by considering well-defined situations. What could be the simplest configurations where flow occurs Suppose we had two parallel plates and a liquid drop squeezed in between. Let us keep the lower plate at rest and move the upper plate at constant velocity in a parallel direction, so that the plate separation distance keeps constant. Near each of the plates, the velocities of the liquid and the plate are equal due to the friction between plate and liquid. Hence a velocity field that describes the stream builds up, (Fig. 15). In the simplest case the velocity is linear in the spatial coordinate perpendicular to the plates. It is a shear flow, as different planes of liquid slide over each other. This is true for a simple as well as for a complex fluid. But what will happen to the mesoscopic structure of a complex fluid How is it affected Is it destroyed or can it even be built up For a review of theories and experiments, see Ref. 122. Let us look into some recent works. 

The close-packed-spheron theory of nuclear structure may be described as a refinement of the shell model and the liquid-drop model in which the geometric consequences of the effectively constant volumes of nucleons (aggregated into spherons) are taken into consideration. The spherons are assigned to concentric layers (mantle, outer core, inner core, innermost core) with use of a packing equation (Eq. I), and the assignment is related to the principal quantum number of the shell model. The theory has been applied in the discussion of the sequence of subsubshells, magic numbers, the proton-neutron ratio, prolate deformation of nuclei, and symmetric and asymmetric fission. 

A two-part series, Designing Oil and Gas Production Systems" by Arnold and Stewart, provides theory and background requirements for selecting two- and three-phase separators. Formulas for liquid-drop velocity, drop diameter, and liquid retention time, as well as step-by-step procedures for selecting both types of separators, are included. Tables provide a means of simplifying vessel sizing calculations. 

C. F. von Weizsacker developed a crude theory of nuclear masses in 1935. The theory takes as its basis the idea that nuclei behave like incompressible uniformly charged liquid drops. How can we account for the variation of nuclear masses We begin by stating that... 

The liquid-drop model was used to model the deformation of sea urchin eggs (Yoneda, 1973). This theory assumes that the tensions in the wall during the compression are uniform and isotropic as is stated by Cole (1932) and Yoneda (1964). However, Hiramoto (1963) suggested that the circumferential tensions are actually up to two times greater than the tensions in the meridian direction. This result suggests that the use of the liquid-drop model may not be appropriate to determine material properties of cells. 

Another theory of the linear energy of the contact line wetting film/bulk liquid drop on a solid surface has been developed by Churaev at al. [478]. These authors also considered both cases of negative and positive line tension. In their interpretation the transition region film/bulk can be presented [478] schematically as shown in Fig. 3.103. The dashed line 1 represents the idealised surface. The real surface is shown for two different cases in case 2 the... 

Theoretical approaches to nucleation go back almost 80 years to the development of Classical Nucleation Theory (CNT) by Volmer and Weber, Becker and Doring and Zeldovich [9,10,17-20]. CNT is an approximate nucleation model based on continuum thermodynamics, which views nucleation embryos as tiny liquid drops of molecular dimension. In CNT, the steady-state nucleation rate /, can be written in the form / a where jS, is the monomer condensation... 

The effective viscosity of a suspension of particles of types other than rigid particles has also been theoretically investigated. Taylor [22] proposed a theory of the electroviscous effect in a suspension of uncharged liquid drops. This theory has been extended to the case of charged liquid drops by Ohshima [17]. Natraj and Chen [23] developed a theory for charged porous spheres, and Allison et al. [24] and Allison and Xin [25] discussed the case of polyelectrolyte-coated particles. 

Heertjes and Kossen [99] present a full discussion of their techniques together with descriptions of the required apparatus and experimental procedure. They considered both the above methods unsuitable for the determination of cos( and proposed a new method, the h-s method. Briefly this consists of determining the height of a drop of liquid placed on top of a cake of the compressed powder previously saturated with the liquid. The theory was presented in an earlier paper [100],... 

The primary result of this analysis is that evaporation occurs only at very low levels of excitation (low velocities of impact). Very hot clusters do not evaporate. They shatter into small pieces. The theory does not exhibit an intermediate regime of cluster fission into two (or three,. ..) roughly equally sized subclusters, as in a liquid drop model.The transition from the evaporative to the shattering regime is a quite abrupt function of the velocity of impact. 

Joseph DD, Belanger J, Beavers GS. Breakup of a liquid drop suddenly exposed to a high-speed airstream. Int J Multiphase Flow 1999 25 1263-1303. Schlichting H. Boundary Layer Theory. New York Me Graw-Hill, 1979. Taylor GI. The intstability of liquid surfaces when accelerated in a direction perpendicular to their planes. Part I. Proc R Soc Lond 1950 A 201 192-196 also in The Scientific Papers of G.I. Taylor. Vol. 3. In Batchelor GK, eds. Cambridge University Press, 1993. 

Bashforth F and Adams JC (1883) An attempt to test the capillary action, Cambridge University Press and Deighton Bell Co., Cambridge Chen P, Kwork DY, Prokop RM, del-Rio 01, Susnar SS and Neumann AW (1998) Axisymmetric drop shape analysis (ADSA) and its applications , in Drops and bubbles in interfacial research, D. Moebius and R. Miller Eds., Studies in Interface Science Series, Vol. 6, Elsevier, Amsterdam Dukhin SS, Kretzschmar G and R. Miller R (1995) Dynamic of adsorption at liquid interfaces. Theory, experiments, applications, D. Moebius and R. Miller Eds., Studies in Interface Science Series, Vol. 1, Elsevier, Amsterdam Joos P (1999) Dynamic Surface Phenomena, VSP, Utrecht, 1999 Kovalchuk VI, Zholkovskij EK, Kragel J, Miller R, Fainerman VB, Wiistneck R, Loglio G and Dukhin SS (2000) Bubble Oscillations in a Closed Cell. J Colloid Interface Sci 224 245-254... 

An objection to this theory is that it has been based on data from the saturated vapors of the fuels and not from dilute air-vapor mixtures. The existence of liquid drops in the dilute fuel mixture drawn through the carburetor, hot intake manifold, and mixed with the hot residual gases in the hot cylinder is doubtful. Also, the fact that such extremely volatile fuels as ethyl ether knock strongly cannot be explained. The fact that some of the most volatile gasolines knock more readily than heavier grades has been attributed to the presence of impurities in the latter which act in an antiknock capacity. 

The mechanisms and data of the fission process have been reviewed recently by Leachman (70). Several different approaches have been used in an effort to explain the asymmetry of the fission process as well as other fission parameters. These approaches include developments of the liquid drop model (50, 51,102), calculations based on dependence of fission barrier penetration on asymmetry (34), the effect of nuclear shells (52, 79, 81), the determinations of the fission mode by level population of the fragments (18, 33, 84), and finally the consideration of quantum states of the fission nucleus at the saddle point (15, 108). All these approaches require a mass formula whereby the masses of the fission fragments far removed from stability may be determined. The lack of an adequate mass formula has hindered the development of a satisfactory theory of fission. The fission process is highly complex and it is not surprising that the present theories fall short of a full explanation. 

The decay of the compound nucleus is assumed in the statistical theory to be independent of the mode of formation of the compound state. The probabilities of decay into different product systems are calculated by analogy with the evaporation of a liquid drop heated to a uniform temperature. The level system of the residual nucleus Y will be excited with a probability, for a given level, dependent on the energy available. Arguments based on the theorem of detailed balancing show that if (o e) (=1/Z)) is the level density at excitation e in the... 

The use of an equation as complex as Eq. (67) requires a lot of numerical calculations so that approximate solutions are very favorable. The first model to describe the adsorption at the surface of a growing drop was derived by Ilkovic in 1938 (107). The boundary conditions were chosen such that the model corresponded to a mercury drop in a polarography experiment. These conditions, however, are not suitable for describing the adsorption of surfactants at a liquid-drop surface. Delahay and coworkers (108, 109) used the theory of Ilkovic and derived an approximation suitable for the description of adsorption kinetics at a growing drop. The relationship was derived only for the initial period of the adsorption process ... 

Danish physicist Aage Bohr (1922- ) and US physicists Benjamin Mottelson (1926- ) and Leo Rainwater (1917-86) combine the liquid-drop and shell models of the nucleus into a single theory. 

But if we are concerned with more complex aspects of nuclear structure, the liquid drop model of the nucleus won t do. Suppose we are interested, for example, in the pattern of stability and instability that governs the collection of nuclear isotopes. Why is there a line of stability about which the stable nuclei are concentrated, with deviation from that line, which is plotted with numbers of protons and numbers of neutrons as axes, indicating the likelihood that the nucleus in question will be unstable Much insight can be gained from a model that treats the nucleons in the nucleus as moving on orbits in an overall potential field. Here, the nucleons are treated as if they were like the electrons in their orbits that surround the nucleus in the atom. Numbers are assigned that are parallels to the familiar quantum numbers of atomic electron theory, and orbits for the nucleons in the nucleus characterized by these quantum numbers are posited. Just... 

The above theory can be extended to electrophoresis of liquid drops [70] and soft particles [71a,b] in salt-free media. The result for liquid drops of viscosity is as follows [70] ... 

A careful account of the problem can be found in Ref. [95]. Ohshima et al. [96] first found a numerical solution of the problem, valid for arbitrary values of the zeta potential or the product Ka. In the same paper, they dealt with the problem of finding the sedimentation potential and the DC conductivity of a suspension of mercury drops. The problems are solved following the lines of the electrophoresis theory of rigid particles previously derived by O Brien and White [18]. The liquid drop is assumed to behave as an ideal conductor, so that electric fields and currents inside the drop are zero, and its surface is equipotential. The main difference between the treatment of the electrophoresis of rigid particles and that of drops is that there is a velocity distribution of the fluid inside the drop, Vj, governed by the Navier-Stokes equation with zero body force (in the case of electrophoresis), and related to the velocity outside the drop, v, by the boundary conditions ... 

This theory of atomic fission was developed by Bohr, who pictured the nucleus as like a liquid drop breaking into two smaller drops under a deform-... 

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