Force-separation curves

This method has been used to obtain force-separation curves for block copolymers adsorbed on planar substrates and subjected to uniaxial compression. Block copolymers are adsorbed from solution onto atomically flat substrates (e.g. mica, quartz) and the force between the plates measured for separations ranging from —0.1 to contact. Further details of the experiments are given in Section 3.8.2. 

When analyzing data from a dissimilar system there are two potentials involved. In Fig. 3 we show theoretical force-separation curves for different pairs of potentials that when multiplied together give the same number. For constant charge systems there is very little difference between the curves produced by the different pairs of potentials. At large separations, where theory is lined to the experimental data to determine the diffuse layer potentials, there is little difference between the constant potential systems. Clearly, there is not a unique pair of diffuse layer potentials that fits the individual experimental force curves. Even when the constant potential interaction fits are considered, any differences between different potential pairs at small separations may be obscured if there is an extra non-DLVO short-range repulsion. For this reason it is necessary to have independently obtained values of the potentials of the materials for comparison. 

FIG. 3 Theoretical force-separation curves for different pairs of potentials that when multiplied together give the same number. The symbols refer to the different potential pairs -35 mV and -70 mV (squares). —50 mV and —50 mV (circles), G —40 mV and -60 mV (triangles). The upper three curves are the constant charge limits, while the lower three arc the constant potential boundary conditions. There is very little difference between the three constant charge curves, and at large separations there is very little difference between the constant potential curves. A Hamaker constant of 2 x 10-20 J was used in the calculations and a background electrolyte of I x I0"1 M NaC l. 

FIG. 5 Force-separation curves taken at G pH 8.2 and an electrolyte concentration of I x 10 4 M KNOi (squares), and pH 3.9 and an electrolyte concentration of I x 10 1 M KNO3 (circles). The dashed lines are theoretical fits to the pH 8.2 data and the full lines to the pH 3.9 data, respectively, where the upper lined curves are the constant charge limits (just visible in the upper left-hand corner of the pH 8.2 data) and the lower curves are the constant potential limits. The pH 8.2 experimental data lies closer to constant charge than constant potential. The fitting parameters are pH 8.2 silica potential -100 mV, alumina potential -34 mV, and Debye length = 30.5 nm pH 3.9 silica potential -25 mV. alumina potential +3 mV, and Debye length = 9.6 nm. A Hamaker constant of 1.8 x l()-20 J was used in the calculations. 

FIG. 7 Force-separation curve taken at the iep of the TiO . pH 5.6. used in this study. The force-separation points have been scaled by the effective radius of the colloid probe. The full curve is the theoretical interaction using the nonretarded Hainaker constant calculated from Lifshitz theory. Ah = 6.1 x I0-20 J. The background electrolyte (KNOv) concentration was 1.0 x 10"4M dm--1. 

Figure S4.6 (a) Force-separation curve and (b) portion of graph shown in part (a). The equilibrium separation of the atoms, ro, occurs when the total force is zero...

By varying the suction pressure, the energy of adhesion could be obtained. In principle it is possible to obtain force separation curves but this has not been achieved, probably because the cells are so compliant. The same technique has been used to measure the adhesion of lecithin vesicles in salt solution. The contact and separation in this case were reversible and the work of adhesion was 13 This compared with the work of adhesion of ordinary red cells in... 

The imaging modes of the AFM can be characterized by the region of the force-separation curve that is being used and by whether the... 

Figure 3.26. Force-separation curve, as in Fig. 3.25, with jump-in and jump-out shown for a cantilever of given spring constant. The dashed line is the restoring force for an oscillating cantilever, rest position r, which always exceeds the attractive force.

On an atomic scale, macroscopic elastic strain is manifested as small changes in the interatomic spacing and the stretching of interatomic bonds. As a consequence, the magnitude of the modulus of elasticity is a measure of the resistance to separation of adjacent atoms, that is, the interatomic bonding forces. Furthermore, this modulus is proportional to the slope of the interatomic force-separation curve (Figure 2.10a) at the equilibrium spacing ... 

Figure 6.7 shows the force-separation curves for materials having both strong and weak interatomic bonds the slope at Tq is indicated for each. 

In order to quantitatively compare the mechanical properties of different phases in the DOPC/ESM/Chol bilayer, high-resolution 2D visual maps of breakthrough forces and elastic moduli, obtained from the analysis of 8192 force-separation curves (128 x 64 array of force curves) using the self-developed code, are shown in Figures 5(a) and 5(c), respertively. 

The products are an oversize (underflow, heavies, sands) and an undersize (overflow, lights, slimes). An intermediate size can also be produced by varying the effective separating force. Separation size maybe defined either as a specific size in the overflow screen analysis, eg, 5% retained on 65 mesh screen or 45% passing 200 mesh screen, or as a d Q, defined as a cut-off or separation size at which 50% of the particles report to the oversize or undersize. The efficiency of a classifier is represented by a performance or partition curve (2,6), similar to that used for screens, which relates the particle size to the percentage of each size in the feed that reports to the underflow. 

As we showed in Chapter 6 (on the modulus), the slope of the interatomic force-distance curve at the equilibrium separation is proportional to Young s modulus E. Interatomic forces typically drop off to negligible values at a distance of separaHon of the atom centres of 2rg. The maximum in the force-distance curve is typically reached at 1.25ro separation, and if the stress applied to the material is sufficient to exceed this maximum force per bond, fracture is bound to occur. We will denote the stress at which this bond rupture takes place by d, the ideal strength a material cannot be stronger than this. From Fig. 9.1... 

Rgure 2.28. Log force vs. arbitrary separation curves for the approach of a glass probe and a droplet from emulsion. The symbols correspond to different levels of cross-linking of PDMS droplets ( ) 50% 45% ( ) 40% (A) 35% (Q) 30%. The background... 

A much easier method has been developed by Padday et al. (1975) (Faraday Trans. I, 71, 1919), which only requires measurement of the maximum force or weight on the rod as it is pulled upwards. It has been shown by using the Laplace equation to generate meniscus profiles that this maximum is stable and quite separate from the critical pull-off force where the meniscus ruptures. A typical force-height curve is shown in Figure 2.24. 

Quantitative evaluation of a force-distance curve in the non-contact range represents a serious experimental problem, since most of the SFM systems give deflection of the cantilever versus the displacement of the sample, while the experimentalists wants to obtain the surface stress (force per unit contact area) versus tip-sample separation. A few prerequisites have to be met in order to convert deflection into stress and displacement into tip-sample separation. First, the point of primary tip-sample contact has to be determined to derive the separation from the measured deflection of the cantilever tip and the displacement of the cantilever base [382]. Second, the deflection can be converted into the force under assumption that the cantilever is a harmonic oscillator with a certain spring constant. Several methods have been developed for calibration of the spring constant [383,384]. Third, the shape of the probe apex as well as its chemical structure has to be characterised. Spherical colloidal particles of known radius (ca. 10 pm) and composition can be used as force probes because they provide more reliable and reproducible data compared to poorly defined SFM tips [385]. 

Fig. 15. Force nomalised by radius as a function of surface separation measured on approach (A) and separation (A). The separation curve is graphically offset by 5mN/m for clarity. The picture shows magnified region of the surface forces curve. The inset shows the entire surface forces curve. Reprinted with permission from Ref. [159]. 2004 American Chemical Society.

The cut-off distance Dc for rigid surfaces is sufficiently so small that it is commonly neglected in the display of force versus separation curves. This cut-off distance may not be neghgible in the presence of absorbed layers of surfactant or polymers, compressed between the surfaces thus it is common to report apparent separation distance on AFM measurements [27]. 

DLVO [2,3] theory estimates the repulsive and attractive force due to the overlap of electric double layers and London-van der Waals force in terms of inter particle distance, respectively. The summation of them gives the total interaction force and can be used for the interpretation of colloid stability in terms of the nature of interaction force-distance curve. If a small interparticle separation (H) is assumed, van der Waals forces for a sphere and substrate can be expressed to... 

The attractive contribution of the typical Lennard-Jones potential varies with the inverse sixth power of the interspecies distance, while the repulsive interactions depend on the inverse twelfth power. By convention, positive energy corresponds to repulsion, and negative energy to attraction. The value of the potential energy for a pair of interacting species at a distance of separation r can be obtained as the integral (i.e., area) under the force—distance curve as shown in Fig. 1.2. 

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