Fluctuations of the Contact Line

The integrals over y extend over the entire system, which has length in the y direction that we denote as L. The difference in energy between this system and the contact energy of the perfect solid is 

We have neglected constant terms and we have used the fact that the average fluctuation in the solid-vapor and solid-liquid tensions is zero. Treating the disorder as a perturbation around the case of the clean surface with a contact line at a = 0, we approximate the value of the surface energy disorder at the contact line, x = X(y), by its value at x = 0. We thus set x = 0 in the argument of w x, y) and approximate 

We must minimize this energy with respect to the two degrees of freedom, h(x, y) and X(y), which determine the shape and position of the fluctuating interface respectively. We first minimize AF with respect to the shape of the fluid-vapor interface, h(x,y) for a fixed but spatially varying contact line, X(y). Thus, 8AF/Sh(x, y) = 0 implies 

Now that we have minimized to find the liquid-vapor interface shape at fixed contact line position, X(y), we evaluate the free energy for this shape (Eq. (4.37)) and minimize to find the optimal value of X(y). We therefore put Eq. (4.37) in Eq. (4.34) for the capillary energy and integrate over the x-coordinate from a = X(y) 0 (since there is already a term proportional to X(y) in the expression for /i(a , y)) to find the change in the interface free energy, AFv, as 

Minimizing the total fi ee energy per unit length, AF = AFs + AF with respect to the Fourier transform of the contact line coordinate, X( ), we find 

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The linear energy of the contact line between two-dimensional phases is only positive. Otherwise the mechanical equilibrium stability condition will be violated. This case is illustrated by examining the fluctuation formation of holes in bilayers. The linear energy of holes in between +6-10 12 J m 1 and +4.5-10 11 J m 1 (+6-10 7 dyn and +4.5-10 6 dyn). 

The non-slip boundary condition is discussed in an excellent paper by Huh and Scriven They take note of the fact that, previous workers seem not to have been well informed by fluid mechanics , in aUuding to the essentially surface chemical analyses of spreading dynamics. Another point they address is that except for very smooth surfaces and non-adsorbing hquids the advancing or receding of the contact line proceeds in a slip-stick and discontinuous fashion a fact which is the focus of attention in current analyses of contact angle hysteresis using the theory of random fluctuations... 

The main message of these lectures is the need to amend the classical hydrodynamic theory by direct inclusion of intermolecular interactions. This is necessary not only in the theory of contact line motion outlined here, but in all mesoscopic hydrodynamic problems, e.g. in fluid mechanics of microdevices, which attracts lately a lot of attention. The specific feature of the contact line problem is the connection between microscopic and macroscopic. The motion in the precursor film can and should be treated more precisely, on the statistical level with due account for fluctuations or directly through molecular dynamics simulations. A challenging problem is matching the microscopic theory with classical hydrodynamics applicable in macroscopic domains away from the immediate vicinity of the contact line. 

Recent analyses of contact angle hysteresis has treated surface heterogeneity in terms of random fluctuations see the excellent review by deGennes The approach is to treat both surface roughness and variations in surface composition as weak fluctuations, i.e. deviations from the ideally smooth surface, du/dy (Fig. 11) and from the difference in solid/liquid and solid/vapor surface energies, — yj - Both fluctuations are considered to be equivalent in perturbing the contact line and are analyzed in terms of their effect on the elastic line energy. 

Fig. 5 Diagram of states for a tethered chain, destaibed by the bond fluctuation model on the simple cubic lattice, for chain length N = 64, in the plane of variables bulk coupling fii, and surface coupling 5. The solid lines show the locatimi of well-deflned maxima in the fluctuation of surface contacts or bead-bead contacts, respectively broken and dotted lines show the locations of less well-pronounced anomalies in these fluctuations. In the hatched region, the precise behavior is still uncertain. The states that compete with each other are desorbed expanded (DE, i.e., a three-dimensional mushroom) adsorbed expanded (AE, i.e., a d = 2 SAW), desorbed collapsed (DC, as in Fig. lb, but tethered to the grafting plane) adsorbed collapsed (AC, a compact but disordered structure with many surface contacts) and various crystalline layered structures (IS). Reprinted with permission from [5]. Copyright 2008, American Institute of Physics...

A close correlation between the formation of this accumulation region and the deposition of particles could be demonstrated. One h othesis suggests that the increase of the local particle concentration is required to initiate the assembly process i) as the local density of particles is very high, the probability of trapping particles increases, and ii) the accumulation of particles contributes to the reduction of particle fluctuation and mobility during the immobilization process and thus increases the trapping efficiency. Indeed, the self-diffusion coefficient of the particle is strongly sensitive to the volume fraction of the suspension which seems to reach a maximum close to the contact line. ... 

Many experiments have been carried out by using this setup the stretching of single DNA molecules, the unfolding of RNA molecules or proteins, and the translocation of molecular motors (Fig. 2). Here we focus our attention on force experiments where mechanical work can be exerted on the molecule and nonequilibrium fluctuations are measured. The most successful studies along this line of research are the stretching of small domain molecules such as RNA [83] or protein motifs [84]. Small RNA domains consist of a few tens of nucleotides folded into a secondary structure that is further stabilized by tertiary interactions. Because an RNA molecule is too small to be manipulated with micron-sized beads, it has to be inserted between molecular handles. These act as polymer spacers that avoid nonspecific interactions between the bead and the molecule as well as the contact between the two beads. 

The home-made heat-flow calorimeter used consisted of a high vacuum line for adsorption measurements applying the volumetric method. This equipment comprised of a Pyrex glass, vacuum system including a sample holder, a dead volume, a dose volume, a U-tube manometer, and a thermostat (Figure 6.3). In the sample holder, the adsorbent (thermostated with 0.1% of temperature fluctuation) is in contact with a chromel-alumel thermocouple included in an amplifier circuit (amplification factor 10), and connected with an x-y plotter [3,31,34,49], The calibration of the calorimeter, that is, the determination of the constant, k, was performed using the data reported in the literature for the adsorption of NH3 at 300 K in a Na-X zeolite [51]. 

Fig. 8.6. Profiles of the lowest (thick line) and of one of the highly excited modes of biaxial (P i,n) and director (P 2,n) fluctuations in the LC heterophase system in contact with (a) ordering and (b) disordering substrates. Dfished lines correspond to the mean-field scalar order parameter. In all cases T —5- Tni-...

Figure 4.2 Scheme summarizing the various stages in the temporal evolution of an unstable liquid film. One can clearly separate the regimes of amplification of surface fluctuations from the subsequent dewetting stages by the time the three-phase contact line is formed—the "touchdown."... 

Although the two contact parameters are sufficient to describe the macrostate of the system and their fluctuations characterize the main pseudophase transition lines, it is often useful to also introduce non-energetic quantities such as the end-to-end distance and the gyration tensor for gaining more detailed structural information of the polymer. For our... 

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