![]() ![]() However, such an approach is somewhat arbitrary and mathematically unjustified. 9 This approach assumed that by plotting the MSD on a log-log scale, one can distinguish between the three modes more easily. Realizing this arbitrariness and shortcomings, other researcher opted for a different approach to define the diffusion mode of MSD which relies on plotting the MSD on a log-log scale. Therefore, extracting D is sensitive to the choice of the fitted region. Moreover, it is known that the MSD converges slowly to the true values which means that the calculated MSD function will be noisy. This issue is more evident in MD simulations where evaluating the MSD at long times is computationally expensive as it requires averaging over a large number of particles and different initial conditions. This arbitrary choice is not justified and can lead to wildly varying and inaccurate estimates of D. In many previous studies, the long-time mode of the MSD was chosen arbitrarily by eyesight to fit to a straight line. However, this procedure suffers from many technical difficulties and ignores the short-time part of the MSD. Typically, this is done by fitting the MSD function to a straight line at long times according to Eq. Extracting an accurate value of D relies heavily on identifying the diffusion regime of the MSD function precisely. In addition, the MSD function of other systems such as nano-confined fluids consists of several modes with nonlinear behavior. The transition from the ballistic mode to the diffusion mode typically goes through a third regime or crossover mode. The second or long-time regime which represents the diffusion mode, the motion is dominated by collisions and particles behave like random walkers where MSD ∼ t. In the ballistic regime, the MSD function can be approximated by a parabola, MSD ∼ t 2. 2,8 The first or the early regime represents the ballistic mode which is a collision-free regime where particles move freely after receiving an initial impulse. For example, the MSD function for many liquids consists of two main regimes (modes). The behavior of the MSD function varies from one system to another but it retains some major features. 6,7 Several experimental methods such as neutron incoherent scattering and nuclear magnetic resonance (NMR) are routinely used to measure the MSD function, in addition to computational methods such as Molecular Dynamics (MD) simulations. 2–5 In addition, diffusion of reaction coordinates can be used to calculate transition rates. The diffusion coefficient is important for modeling transport processes in real liquids and other systems such as biological systems and nano-confined geometries. The results of our fits are in good agreement with the experimentally reported values. ![]() In addition, we applied our algorithm to extract D for water based on Molecular Dynamics (MD) simulations. We tested our algorithm using numerical experiments, and our fits described the data remarkably well. The novelty of this approach lies in its ability to find the breakpoints which separate different modes of motion. The algorithm fits the MSD to a continuous piece-wise function and predicts all the coefficients in the model including the breakpoints. The algorithm presented here achieves two major goals a more accurate estimation of D as well as extracting information about the short time behavior. In this work, we present a new optimal and robust nonlinear regression model capable of fitting the MSD function with different regimes corresponding to different time scales. This paper is concerned with fitting the mean-square displacement (MSD) function, and extract reliable and accurate values for the diffusion coefficient D. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |