Stochastic disks that roll

Abstract

We study a model of rolling particles subject to stochastic fluctuations, which may be relevant in systems of nano-or micro-scale particles where rolling is an approximation for strong static friction. We consider the simplest possible non-trivial system: a linear polymer of three of discs constrained to remain in contact, and immersed in an equilibrium heat bath so the internal angle of the polymer changes due to stochastic fluctuations. We compare two cases: one where the discs can slide relative to each other, and the other where they are constrained to roll, like gears. Starting from the Langevin equations with arbitrary linear velocity constraints, we use formal homogenization theory to derive the overdamped equations that describe the process in configuration space only. The resulting dynamics have the formal structure of a Brownian motion on a Riemannian or subRiemannian manifold, depending on if the velocity constraints are holonomic or non-holonomic. We use this to compute the trimer's equilibrium distribution both with, and without, the rolling constraints. Surprisingly, the two distributions are different. We suggest two possible interpretations of this result: either (i) dry friction (or other dissipative, nonequilibrium forces) changes basic thermodynamic quantities like the free energy of a system, a statement that could be tested experimentally, or (ii) as a lesson in modeling rolling or friction more generally as a velocity constraint when stochastic fluctuations are present. In the latter case, we speculate there could be a "roughness" entropy whose inclusion as an effective force could compensate the constraint and preserve classical Boltzmann statistics. Regardless of the interpretation, our calculation shows the word "rolling" must be used with care when stochastic fluctuations are present.Particles that live on the nano-or microscale commonly have short-ranged interactions, so their surfaces come close enough that surface frictional effects may be important. For example, recent experiments and simulations have shown that tangential frictional forces between rough, and otherwise stochastic particles, are probably the origin of the shear-thickening behaviour of many materials [1,2]. Other studies demonstrate that sticky tethers attached to particle surfaces can change their dynamics [3,4]. Since one promising method of creating colloids with programmable interactions is to coat them with strands of DNA [5][6][7][8], which could impede their relative sliding, this could have major implications for their assembly pathways and hence structures that can be formed by selfassembly. On these scales it is extremely difficult to measure the particles' rotational degrees of freedom, so one must resort to indirect methods to determine whether tangential frictional forces are present [9,10]. Therefore, it would be highly desirable to find a simpler way to quantify these forces, via macroscopic measurements of spatial positions only.While the mascroscopic effect of dry friction has been studied in detail i...

Modeling the relative dynamics of DNA-coated colloids

“…an intriguing possibility is whether this unusual growth is aided in part by the relative dynamics of the DNAbased interactions, such as a preference for rolling [41]. (Note that reaction-limited kinetics, another possible effect of DNA-induced interactions, can also lead to lowcoordination open structures [22].…”

“…If particles prefer to move in ways that minimize the amount of surface-surface rubbing, then dynamically they may behave like gears, and may get stuck for long times in more open arrangements where the number of contacts is less than the isostatic number needed to stabilize spheres mechanically. It has been observed in recent experiments that crystals of DNA-coated colloids like to grow as open, ordered structures, such as in a diamond lattice, transitioning only once they are big enough to more compact structures with more contacts and lower energy [8]; an intriguing possibility is whether this unusual growth is aided in part by the relative dynamics of the DNAbased interactions, such as a preference for rolling [41]. (Note that reaction-limited kinetics, another possible effect of DNA-induced interactions, can also lead to lowcoordination open structures [22].…”

The Nanocaterpillar's Random Walk: Diffusion With Ligand-Receptor Contacts

“…For example, leg flexibility and bond lifetimes control the average mobility of the particle [19,23,24], and differences in both parameters can be harvested to detect infected cells [25][26][27] or prevent viral infections [28]. Leg density affects how DNA-coated colloids nucleate and grow into crystals [29,30] and governs the long-range alignment of crystals [31][32][33]. Overall, microscopic details underlie a variety of large-scale modes of * sophie@marbach.fr q on q off Overview of nanocaterpillars.…”

Mass changes the diffusion coefficient of particles with ligand-receptor contacts in the overdamped limit

“…( 3) is m → m + m f /2 where m f is the mass of the displaced volume of fluid [2,32]. We remark that compared to a starting point with overdamped equations for both the leg and the particle, here it is not necessary to project the unbound dynamics to obtain the bound dynamics [28,33]; Newton's law is sufficient. The Langevin dynamics in Eq.…”

“…Improvements to our model could include fluid memory kernels to investigate the decay of velocity autocorrelation functions [2] or inhomogeneities in the lateral direction to probe inertial effects on the subdiffusion regime [38]. For DNA-coated colloids, one could envision that such inertia-modified dynamics could also impact collective properties such as crystallographic alignment into self-assembled structures [33,51]. Understanding the dynamics of such complex micronscale particles is a key step to pave the way towards controlled design at the microscale, e.g.…”