Surface forces summing

Interactions between macromolecules (protems, lipids, DNA,.. . ) or biological structures (e.g. membranes) are considerably more complex than the interactions described m the two preceding paragraphs. The sum of all biological mteractions at the molecular level is the basis of the complex mechanisms of life. In addition to computer simulations, direct force measurements [98], especially the surface forces apparatus, represent an invaluable tool to help understand the molecular interactions in biological systems. 

Here Tq is the unit vector perpendicular to the z-axis, and r is its distance to the point q. From this equation it follows that the sum of the two true forces represents the centripetal force. In other words, the distribution of the attraction and surface forces per unit volume ... 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

Fig.1.a Variation of the interaction force between a flat surface and an isolated atom in the near field of the surface. The dot line corresponds to the lateral displacement of the atom by one interatomic distance. The grey line indicates the force profile sensed by the atom as it moves parallel to the surface plane, b Scheme of a SFM probe a sharp tip mounted on a cantilever. The interaction force Fi=Fs+Fd is a sum of many interatomic interactions, where Fs is the surface force and force Fd results from the sample deformation. The interaction force is balanced by force Fc due to the cantilever bending... 

From the above discussion it is logical to assume that adsorbed layer thickness T, calculated as the ratio of VJS, is the maximum distance of the influence of the surface forces. The sum of the excess adsorption value and the product of the equilibrium concentration and adsorbed layer volume represent the total amount of the adsorbate in that layer for any given equilibrium concentration... 

Surface forces acting between mesoscopic or macroscopic bodies can be described by summing up pairwise interatomic (intermolecular) forces for each participating atomic (molecular) pairs within the interacting bodies. Semiempirical... 

The role of the medium, in which contacting and pull-off are performed, has been mentioned but not considered so far. However, the surroundings obviously influence surface forces, e.g., via effective polarizability effects (essentially multibody interactions e.g., by the presence of a third atom and its influence via instantaneous polarizability effects). These effects can become noticeable in condensed media (liquids) when the pairwise additivity of forces can essentially break down. One solution to this problem is given by the quantum field theory of Lifshitz, which has been simplified by Israelachvili [6]. The interaction is expressed by the (frequency-dependent) dielectric constants and refractive indices of the contacting macroscopic bodies (labeled by 1 and 2) and the medium (labeled by 3). The value of the Hamaker constant Atota 1 is considered as the sum of a term at zero frequency (v =0, dipole-dipole and dipole-induced dipole forces) and London dispersion forces (at positive frequencies, v >0). 

It is customarily assumed that the overall particle-particle interaction can be quantified by a net surface force, which is the sum of a number of independent forces. The most often considered force components are those due to the electrodynamic or van der Waals interactions, the electrostatic double-layer interaction, and other non-DLVO interactions. The first two interactions form the basis of the celebrated Derjaguin-Landau-Verwey-Overbeek (DLVO) theory on colloid stability and coagulation. The non-DLVO forces are usually determined by subtracting the DLVO forces from the experimental data. Therefore, precise prediction of DLVO forces is also critical to the determination of the non-DLVO forces. The surface force apparatus and atomic force microscopy (AFM) have been used to successfully quantify these interaction forces and have revealed important information about the surface force components. This chapter focuses on improved predictions for DLVO forces between colloid and nano-sized particles. The force data obtained with AFM tips are used to illustrate limits of the renowned Derjaguin approximation when applied to surfaces with nano-sized radii of curvature. 

Adhesion of individual particles to each other or to solid surfaces is controlled by the competition between volume and surface forces (A.5 in Table 1). In order to cause adhesion, certain criteria must be fulfilled. The most important of all is that any environmental forces (e.g. gravity, inertia, drag, etc.) must be smaller than the attraction forces between the adhering partners. According to Figure 8 and equation (1), the ratio between all binding forces, 5/(jr), and the sum of the active components of the environmental forces, Fjy x), involved is a measure for the adhesion tendency... 

Another difficulty arises from the nonlinear nature of the calculation, which may often cause the surface of sum of weighted squares of residuals to have multiple minima as a function of the force constants, so that it may be possible to converge onto several different minima by starting from different trial force fields in the refinement. This can be particularly troublesome when the data are only just sufficient to determine the force field, so that the normal equations are somewhat ill-conditioned. The nature of the calculation is reminiscent of S. D. 

The left-hand side is just the time rate of change of linear momentum of all the fluid within the specified material control volume. The first term on the right-hand side is the net body force that is due to gravity (other types of body forces are not considered in this book). The second term is the net surface force, with the local surface force per unit area being symbolically represented by the vector t. We call t the stress vector. It is the vector sum of all surface-force contributions per unit area acting at a point on the surface of Vm(t). 

Fig. 10.1.4. The liquid may spread freely over the surface, or it may remain as a drop with a specific angle of contact with the solid surface. Denote this static contact angle by 6. There must be a force component associated with the liquid-gas surface tension (t that acts parallel to the surface and whose magnitude is a cos 0. If the drop is to remain in static equilibrium without moving along the surface, it has to be balanced by other forces that act at the contact line, which is the line delimiting the portion of the surface wetted by the liquid, for example, a circle. It is assumed that the surface forces can be represented by surface tensions associated with the solid-gas and solid-liquid interfaces that act along the surface, and tr i, respectively. Setting the sum of the forces in the plane of the surface equal to zero, we have...

To find the z component of this force, we sum the z components of the four surface forces on the edges to find... 

Figure 5 demonstrates shear stress variation for k = 10 flow inside 5.4 nm channel as the sum of its kinetic and virial contributions. Shear stress is constant in the bulk and shows variations within 0.34 nm ( s) from wall. Shear stress is defined by the kinetic term in most of the domain, while the surface virial increases starting from one sigma distance from the wall till the gas density reduces to zero due to the impenetrable wall zone. Therefore, near-wall spatial variations of shear stress are induced by the surface force field effects and the wall motion. 

The DLVO model (named after its principal creators, Deqaguin, Landau, Verwey and Overbeek) is the most widely used to describe inter-particle surface force potential (1,2). It assumes that the total inter-particle potential is the sum of an attractive van der Waals force and a repulsive double-layer force. The repulsive force due to the double-layer coulombic interaction between equal spheres separated by a distance D generates a positive potential energy Vr. If the radius r of the spheres is large compared to the double-layer thickness 1/k (Kr l, with K the Debye-Hiickel parameter), Fr is described approximately by ... 

Fluid flow past a surface or boundary leads to surface forces acting on it. These surface forces depend on the rate at which fluid is strained by the velocity field. A stress tensor with nine components is used to describe the surface forces on a fluid element. The tangential component of the surface forces with respect to the boundary is known as shear stress. The nature or origin of shear stress depends on the nature of flow, i. e., laminar or turbulent. The stress components for a laminar flow are functions of the viscosity of the fluid and are known as viscous stresses. The turbulent flow has additional contributions known as Reynolds stresses due to velocity fluctuation, i. e., the stresses of a laminar flow are increased by additional stresses known as apparent or Reynolds stresses. Hence, the total shear stresses for a turbulent flow are the sum of viscous stresses and apparent stresses. In a turbulent flow, the apparent stresses may outweigh the viscous conponents. 

On a macroscopic scale, the spontaneous polarization vector in the optically active phase spirals about an axis perpendicular to the smectic layers (Fig. 20), and sums to zero. This macroscopic cancellation of the polarization vectors can be avoided if the helical structure is unwound by surface forces, by an applied field, or by pitch compensation with an oppositely handed dopant. The surface stabilized ferroelectric liquid crystal display utilizes this structure and uses coupling between the electric field and the spontaneous polarization of the smectic C phase. The device uses a smectic C liquid crystal material in the so-called bookshelf structure shown in Fig. 21a. This device structure was fabricated by shearing thin (about 2 i,m) layers of liquid crystal in the... 

If two smooth elastic bodies are brought into contact, say a deformable rubber and a rigid plane, the surface forces create an adhesive function and work must be done to separate the bodies. " Rather than summing the surface forces across the interface to account for the adhesion, the net surface energy change on creating the interface may be computed as... 

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