What is the physical significance of the damping function in calculation of vdW interaction energy?

In order to take care for the singularity (also removing the double counting of exchange-correlation) obtained with this expression,
$$E\_\mathrm{vdW} = \frac{C^6}{R^6}$$


many type of damping functions are utilized. Such as,


$$E\_\mathrm{vdW} = \frac{C^6}{R^6} \cdot f\_\mathrm{famp}(r)$$


One such damping function is, (Grimme type, Fermi function)


$$f\_\mathrm{damp}(r) = \frac{s\_6}{1+\exp\left[ -B\left(\frac{r^{AB}}{s\_r R^{AB}\_\mathrm{vdW}}\right)\right]}$$


when, $r$ = interatomic distance, $R^{AB}\_\mathrm{vdW}$ or (R) is summation of vdW radius of two atoms $s\_6$ and $s\_r$ are two parameters.
$s\_6$ depend upon the XC functional and $s\_r$ is another scaling factor.
$B$ is the exponent.


Literature$^{[1]}$ suggests that value for $B$ is found by setting $f\_\mathrm{damp} = 0.99$ when $r = 1.2 \cdot R$.


Also, the value of these parameters can also be found from [here](https://www.chemie.uni-bonn.de/pctc/mulliken-center/software/dft-d3/functionals).


To my knowledge, these values are optimized for each functional.
Is there any physical meaning to these values of the parameters?
Actually, I am more interested in the $s\_r$ value and if this has any physical significance...


[1] J Chem Phys, 116, 515(2002)


