Jamming for nematic deposition in the presence of impurities

The deposition of one-dimensional objects (such as polymers) on a one-dimensional lattice with the presence of impurities is studied in order to find saturation conditions in what is known as jamming. Over a critical concentration of $k$-mers (polymers of length $k$), no further depositions are possible. Five different nematic (directional) depositions are considered: baseline, irreversible, configurational, loose-packing, and close-packing. Correspondingly, five jamming functions are found, and their dependencies on the length of the lattice, $L$, the concentration of impurities, $p=M/L$ (where $M$ is the number of one-dimensional impurities), and the length of the $k$-mer ($k$) are established. In parallel, numeric simulations are performed to compare with the theoretical results. The emphasis is on trimers ($k=3$) and $p$ in the range [0.01,0.15], however other related cases are also considered and reported.

Figure 1

Two possible distributions for four one-site impurities on an $L=34$ one-dimensional lattice. (a) A random distribution with one segment of nil length (two impurities touching each other); (b) an equally spaced distribution with all segments of length 6.

Figure 2

Segment length distribution for $L=2000, M=200$ ($p=0.10$); average values after 1000 initializations.

Figure 3

Average segment length as a function of impurity content for $L=2000$ (solid triangles), compared to the average segment length for a system where $L\to \infty$ (open squares).

Figure 4

Unoccupied portion of the ODL by means of blocking analysis: ${\upsilon }_p^k$ for $k=2$ and 3 after simulations for $L=2000$.

Figure 5

All possible 21 depositions of two trimers within a segment of length 11. Just three of them lead to jamming; they are marked by an external rectangle. All other 18 depositions accept one more trimer.

Figure 6

Jamming concentrations ${\gamma }_{\ell }^3$ as labeled in the inset where indices are omitted. Depositions correspond to trimers over segments of increasing length up to $\ell =87$.

Figure 7

Distribution for the jamming values for RSD after 50 000 depositions ($x=3n/\ell$, where $n$ is the number of saturating trimers) over a segment of length $L=\ell =80$ (no impurities).

Figure 8

Jamming function ${\sigma }_{\ell }^k$ for $k=2$, 3, and 4 for the baseline deposition as functions of the segment length.

Figure 9

Jamming function for the full lattice for trimer irreversible deposition. Red circles are obtained from Eq. (32) in the text, corresponding to a projected result from previously simulated depositions on segments. Black squares correspond to direct irreversible deposition of trimers over a whole lattice with $L=10000$ for impurity concentration $p$.

Figure 10

Projected jamming functions for trimers on ODLs with impurity concentration $p$ for the five depositions indicated in the inset.