Threshold force

All our discussion has assumed that our branched structures are robust. But they are in fact subjected to significant hydrodynamic tensions. Consider a star polymer under a current J largerthan Jcl. Then we shall meet situations where one (or a few) arm(s) has entered the tube, while the central nodule is still stuck at the entry as in Fig. 3b. If is the length of tube which has been invaded by the arm(as in Sect. 2.3), the hydrodynamic force qv/, may be large, and can possibly reach the threshold force xm for rupture near the nodule. This leads to a critical value m ... 

Similar ideas can be used for statistically branched polymers let us, for instance, consider the case of large distances b between branch points, i.e. strong confinement as defined by the inequality (20). Then the tube (of cross section D2) can be subdivided into (D/ )2 subtubes , and the chain, in each of them, is subjected to the tension r[v . When the tension equals the threshold force, this leads again to Eq.(34). Here, the current threshold Jc is much larger than Jcl, and rupture may often occur. To avoid this, we need to operate in situations of mild compression, i.e. with D/Rg not too small. 

It is difficult to see how the force transmission would degrade as the asperity topography flattens out with wear. To a first-order approximation, the volume of material removed should be linearly proportional to the force times the bearing area (F x A). As the asperity peaks are smoothed out to a planar surface, the individual forces that were exerted by the fewer peaks is reduced, but in the same proportion as the increase in the bearing area. Unless one can justify a threshold force (vis-a-vis activation energy), linearity should hold and the material removal rate (MRR) should not decrease as the individual asperities are worn down. 

For weakly wetted systems, the mobilization process appears to occur as follows. The receding surface first begins to move through the throat increasing the pressure drop across the advancing surface then, a jump of the advancing surface (which requires a significant threshold force) rapidly follows. This behavior could not have been identified from capillary-type experiments. 

To achieve the practical implementation of self-reporting materials in real-world products, several key factors have yet to be addressed. For example, it has to be made sure that the self-reporting mechanisms are stable over long periods of time and that they do not produce false positive or false negative results. Moreover, in many cases the threshold forces to trigger mechanochromic events in materials have not yet been determined in detail, as they depend on a complex interplay of the properties of the mechanochromic reporter, the mechanical properties of the material, and the efficiency of force transduction from the material onto the mechanophores, proteins, microcapsules, etc. Given the current vibrant dynamic of the research field, it can be expected that such questions will be addressed in future research. This will also require the refinement of known mechanophores and the development of new mechanochromic systems. 

We find that break off is impossible to trigger below a threshold force. The line remains fixed for angles 6 < Or. 

Under these conditions, it is straightforward to evaluate the threshold force per unit length required for the line tO advance, which is given (at the macroscopic level) by the expression... 

Near the Lifshitz line thermal composition fluctuations are expected to become strong over a larger temperature range because of the reduced smTace energy (c2 a I2 = 0), leading to a lower threshold force for thermal fluctuations. On a more abstract level this effect can also be interpreted in terms of a larger upper critical dimension Du = 8 beyond which thermal fluctuations become irrelevant, and Gi is twice as large as for ordinary binary polymer blends [86]. 

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