Scaling of Forces

B2. Barnett, P. G., The scaling of forced convection boiling heat transfer, AEEW-R.134 (1963). 

Eisler, H. (1962), Subjective Scale of Force for a Large Muscle Group, Journal of Experimental Psychol., Vol. 64, No. 3, pp. 253-257. 

AFM force measurements can be done with a range of surfaces. Conventional AFM tips, however, should not be used for quantitative interfacial force measurements because quantitative comparison with theoretical predictions requires that the radius of the approaching probe be much greater than the separation distance. The latter point has often been overlooked in surface force measurements with standard AFM tips. The coUoid-probe AFM method > > is appropriate for quantitative surface force measurements. A colloidal (usually silica) microsphere attached to the end of the AFM cantilever provides a well-defined, mathematically tractable sphere-vs-flat geometry for the scaling of forces, and allows the use of different colloid materials, or the surface modification of colloid spheres, for investigating interactions between surfaces with various physicochemical properties. 

Due to the similarity of the flow profiles, most of these passive, continuous-flow schemes can, in principle, also be adopted for centrifugally driven flows. In addition, the availability of the Coriolis pseudo force/c (4) offers an intrinsic means for the generation transversal flow components, even in straight radial channels exhibiting a constant cross section (Fig. 8). Due to the scaling of forces //c (13), the Coriolis-force induced mixing is... 

The most important validation of the model will be the comparison of the experimental and numerical force-displacement curves. If the numerical model is simulating the behavior of the knitted stmcture correctly, then the shapes of these curves should be very similar and the only difference should be the scale of force since the numerical simulation uses fewer filaments. Visual checks can also be used, e.g., comparison of loop height and width or wale/course density given as number of vertical/horizontal columns/rows of loops per cm for both the experimental and numerical case. 

Figure Bl.20.7. The solvation force of ethanol between mica surface. The inset shows the fiill scale of the experimental data. With pennission from [75].

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

Abstract. Molecular dynamics (MD) simulations of proteins provide descriptions of atomic motions, which allow to relate observable properties of proteins to microscopic processes. Unfortunately, such MD simulations require an enormous amount of computer time and, therefore, are limited to time scales of nanoseconds. We describe first a fast multiple time step structure adapted multipole method (FA-MUSAMM) to speed up the evaluation of the computationally most demanding Coulomb interactions in solvated protein models, secondly an application of this method aiming at a microscopic understanding of single molecule atomic force microscopy experiments, and, thirdly, a new method to predict slow conformational motions at microsecond time scales. 

Fig. 5. Theory vs. experiment rupture forces computed from rupture simulations at various time scales (various pulling velocities Vcant) ranging from one nanosecond (vcant = 0.015 A/ps) to 40 picoscconds (vcant = 0.375 A/ps) (black circles) compare well with the experimental value (open diamond) when extrapolated linearly (dashed line) to the experimental time scale of milliseconds.

Time-reversible energy conserving methods can be obtained by appropriate modifications to the (time-reversible) midpoint method. Two such modifications are (i) scaling of the force field by a scalar such that total energy... 

To separate the non-bonded forces into near, medium, and far zones, pair distance separations are used for the van der Waals forces, and box separations are used for the electrostatic forces in the Fast Multipole Method,[24] since the box separation is a more convenient breakup in the Fast Multipole Method (FMM). Using these subdivisions of the force, the propagator can be factorized according to the different intrinsic time scales of the various components of the force. This approach can be used for other complex systems involving long range forces. 

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