Semi-infinite system

Although the properties of specific polymer/wall systems are no longer accessible, the various phase transitions of polymers in confined geometries can be treated (Fig. 1). For semi-infinite systems two distinct phase transitions occur for volume fraction 0 = 0 and chain length N oo, namely collapse in the bulk (at the theta-temperature 6 [26,27]) and adsorp-... 

In cases when the two surfaces are non-equivalent (e.g., an attractive substrate on one side, an air on the other side), similar to the problem of a semi-infinite system in contact with a wall, wetting can also occur (the term dewetting appHes if the homogeneous film breaks up upon cooHng into droplets). We consider adsorption of chains only in the case where all monomers experience the same interaction energy with the surface. An important alternative case occurs for chains that are end-grafted at the walls polymer brushes which may also undergo collapse transition when the solvent quality deteriorates. Simulation of polymer brushes has been reviewed recently [9,29] and will not be considered here. 

As an adsorption geometry one considers a semi-infinite system with an impenetrable wall at z = 0, such that monomer positions are restricted to the positive half-space z > 0. At the wall acts a short-range attractive potential, either as a square well... 

The last quantity that we discuss is the mean repulsive force / exerted on the wall. For a single chain this is defined taking the derivative of the logarithm of the chain partition function with respect to the position of the wall (in the —z direction). In the case of a semi-infinite system exposed to a dilute solution of polymer chains at polymer density one can equate the pressure on the wall to the pressure in the bulk which is simply given by the ideal gas law The conclusion then is that [74]... 

H. W. Diehl, S. Dietrich. Field theoretical approach to static critical phenomena in semi-infinite systems. Z Phys B 42 65, 1981. 

As an example of the solution of Eq. (42), Lapidus and Amundson (L3) and Robinson (R4) considered the closed-open (semi-infinite) system and obtained the solution ... 

In all of the above cases we have in fact assumed that end rarefactions can be neglected, ie, we have considered semi-infinite systems. In the real world rarefactions cannot be neglected, particularly for small expl/metal configurations. Consequently we will examine the effects of rarefactions for cylindrical expl/metal geometries Fig 12 (from Ref 24) shows that observed fragment velocities near the ends of a cylindrical... 

In Section 4.2.3 we described application of the method of superposition to infinite and semi-infinite systems. The method can also be applied, in principle, to finite systems, but it often becomes unwieldy (see Crank s discussion of the reflection method [2]). 

Let us consider now a nanosystem coupled to a semi-infinite lead (Fig. 3). The direct matrix inversion can not be performed in this case. The spectrum of a semi-infinite system is continuous. We should transform the expression (26) into some other form. 

If we allow diffusion to continue until t=t2>t, then under the same assumptions of a semi-infinite system, the mass transfer during At = t2—ti is... 

Fig. 14. Schematic phase diagram of the surface of a semi-infinite block copolymer (a) and a related system in a thin film geometry, assuming two walls a distance D apart, at which both the same surface field Hx acts b.In the semi-infinite system for the profile of...

Fig. 28. Averaged order parameter profiles cf>av(Z,x) plotted vs the scaled distance Z=z/2 b from the left wall at z=0 for four different scaled times T after the quench as indicated, for a scaled distance D =D/2 b=60. Choosing a rescaled distance L /2 b=600, and a discretization AX=1.5, Ax=0.05, the resulting equations are solved by the cell-dynamics method. The results shown are for parameters h1=y=4, g =-4, and averaged over 2000 independent initial conditions, corresponding to random fluctuations in a state with J( )av(Z,0)dZ=0. The parameters Iq and g were chosen such that both walls prefer A but one is still in the non-wet region of the equilibrium surface phase diagram of the corresponding semi-infinite system. From Puri and Binder [145]...

Norton et al. [80] also extract dfsbare(( )o)/d( )Q as function of ( )q from their data, using both the mean field theory for semi-infinite systems [11] and a different method based on the Gibbs adsorption equation [80]. They find from both analyses that - dfbare(( )0)/d(t>0 stays constant for ( )q<0.6 and steeply rises when c])0—>1. Again the caveat must be mentioned that possible effects due to the finite thickness of the total film have not been considered in this work. 

In slab calculations, a finite number of layers mimicks the semi-infinite system, with a two-dimensional (2D) translational periodicity. A minimal thickness dmin is required, so that the layers in the slab centre display bulk characteristics. Practically speaking, dm-,n should be at least equal to twice the damping length of surface relaxation effects, which depend upon the surface orientation. In plane wave codes, the slab is periodically repeated... 

Fig. 5 Iso-density map of an excess electron state on TiO2(110). Titanium and oxygen atoms are marked by small and large circles, respectively, and the semi-infinite system is represented by a slab with vacuum above and below and lateral periodic boundary conditions (from Ref. 157).

In addition, the surface orientation h is not always sufficient to fully characterize a semi-infinite system, especially when various terminations may be produced. In the rutile structure, for example, the bulk repeat unit in the (110) direction is made of three layers of O and (MO)2 composition, and there exist three chemically inequivalent terminations, which expose a single oxygen layer (0/(M0)2/0 sequence), two oxygen layers (0/0/(M0)2 sequence) or one mixed cation-oxygen layer ((M0)2/0/0 sequence). Only in the first case, the repeat unit bears no dipole mo-... 

Real polymer mixtures studied here do not form semi-infinite systems but rather they are confined in thin layers bounded by two surfaces. For relatively thick films (for a critical thickness evaluation see Sect. 3.2) the equilibrium profile ( >(z) of the whole film is described separately for each of the two surfaces allowing their independent characterization. This is based on the assumption that the profile ( >(z), describing the segregation to the respective surface, is in equilibrium with the plateau value of < > in the region adjacent to this surface. This approach was justified by theoretical [180, 181] and experimental [182] works on the dynamics of surface segregation, and is used here to focus on phenomena occurring near a single surface. 

Consider now such a semi-infinite system, confined on one side by an infinite planar surface, and assume that a given potential I .s is imposed on this surface. The interior bulk potential is denoted g. Having h.s implies that the mobile... 

In sect. 2, we have summarized the general theory of phase transitions with an emphasis on low-dimensional phenomena, which are relevant in surface physics, where a surface acts as a substrate on which a two-dimensional adsorbed layer may undergo phase transitions. In the present section, we consider a different class of surface phase transitions wc assume e.g. a semi-infinite system which may undergo a phase transition in the bulk and ask how the phenomena near the transition are locally modified near the surface, sect. 3.1 considers a bulk transition of second order, while sections 3.2 and 3.4-3.6 consider bulk transitions of first order. In this context, a closer look at the roughening transitions of interfaces is necessary (sect. 3.3). Since all these phenomena have been extensively reviewed recently, we shall be very brief and only try to put the phenomena in perspective. 

The theory of wetting phenomena has been extensively reviewed recently (Sullivan and Telo da Gama, 1986 Dietrich, 1988) we present here a very brief introduction only. We return to the free energy of a semi-infinite system [eqs. (219), (220)] and rescale the parameters such (Schmidt and Binder, 1987) that all the parameters of the bulk free energy density are absorbed in the rescaled bulk field fi, order parameter n(Z) and rescaled distance Z,... 

Fig. 53. Schematic isotherms (density p versus chemical potential pi) corresponding to the gas-liquid condensation in capillaries of thickness D, for the case without (a) and with (b) prewetting, and adsorption isotherm (c) for a semi-infinite system, where the surface excess density pjs is plotted vs. pi. Full curves in (a) and (b) plot the density p vs. pi for a bulk system, phase coexistence occurs there between p,p, (bulk gas) and pn, (bulk liquid), while in the capillary due to the adsorption of fluid at the walls the transition is shifted from paKX to a smaller value rc(D, 7) (with pic(7>, T) 1 /D, the Kelvin equation ), and the density jump (from ps D) to pt D)) is reduced. Note also that in the ease where a semi-infinite system exhibits a first-order wetting transition 7W, for 7 > 7W one may cross a line of (first-order) prewetting transitions (fig. 54) where the density in the capillary jumps from p to p>+ or in the semi-infinite geometry, the surface excess density jumps from p to p +, cf. (c), which means that a transition occurs from a thin adsorbed liquid film to a thick adsorbed film. As pi the thickness of the adsorhed liquid film in the semi-infinite...

Fig. 54. Schematic phase diagrams for wetting and capillary condensation in the plane of variables temperature and chemical potential difference, (a) Refers to a case in which the semi-infinite system at gas-liquid condensation (ftaKX — d = 0) undergoes a second-order wetting transition at T = 7V The dash-dotted curves show the first-order (gas-liquid) capillary condensation at p = jt(I), T) which ends at a capillary critical point T v, for two choices of the thickness D. For all finite D the wetting transition then is rounded off. (b), (c) refer to a case where a first-order wetting transition exists, which means that ps remains finite as T - T and there jumps discontinuous towards infinity. Then for /iaKX - /i > 0 a transition may occur during which the thickness of the layer condensed at the wall(s) jumps from a small value to a larger value ( prewelting ). For thick capillaries, this transition also exists (c) but not for thin capillaries because then /Jcnn - (D,T) simply is loo large.

It is to be emphasised that property (v), which can be obtained similarly for the semi-infinite system (—c , 0), does establish the existence of the Gibbs state (i.e., the thermodynamic limit of the finite-volume canonical equilibrium state). Property (vi) establishes, on the other hand, the absence of phase transitions (as is well-known). 

HENISCH A) We dealt with a semi-infinite system. On one side, therefore, the boundary conditions are those which characterize the undisturbed bulk material, on the other side, our X=0, we assume Jp/J = 1 and F = field = 0. The first implies maximum injection Cjust to find out what happens then), the second comes from p-n junction theory. There is a Iway s a point where F = 0 when a forward voltage is applied to a junction. We simply found it convenient to call that point X=D. Ng and Pq had to be determined by reiterative calculations. 

The role of overlap in Pauli repulsion will be stressed. The extended Hiickel method will be applied to clusters as well as semi-infinite systems. 

Then the relevant theoretical concepts to be used in bonding to semi-infinite systems with a continuous distribution of energy levels, instead of discrete levels as in clusters, will be introduced. 

A metal surface is part of a semi-infinite system. Apart from changes in electronic structure and electrostatics, a major difference between a finite and infinite system is, that there is no local conservation of the total number of electrons. The local number of electrons is determined by the requirement that for a system at equilibrium the Fermi-level energy, i.e., the energy of the Highest occupied molecular orbital is the same throughout the system. 

In semi-infinite systems it is required that the Fermi level does not change and this allows the development of convenient closed expressions for changes in energy due to surface-formation or chemisorption. Such expressions have been derived by Koutecky [47], Grimley [48], Schrieffer [49], and others [50]. 

Equation (2) is correct for infinite systems where there is no contribution of energy of the surface to the total energy of the system. For the semi-infinite system having a limiting surface, under assumption that the volume remains unchanged after the phase transition, equality (2) transforms into the equation ... 

To obtain the dependence between melting temperatures T , in an infinite system and T in a semi-infinite system, we substitute (6) for (5) considering (3)... 

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