Schematic phase diagram for tethered chains of finite length in the space of field variables and (see Sec. III for the formal definition). As discussed in the text, increasing values of and correspond to increasingly attractive surface- and monomer-monomer interactions, respectively. The lines indicate transitions between states identified by the following abbreviations: DE for desorbed extended (mushroom), AE for adsorbed extended (pancake), DC for desorbed compact, AC for adsorbed compact, and LS for layered states. The solid lines indicate transitions that are expected to become true phase transitions in the limit of infinite chain length. The dashed lines represent structural transitions observed for finite size chains only. The ending of the dashed lines indicates that, in simulations, these structural transitions can no longer be uniquely identified in regions of field parameters, where several transitions compete with each other.
Ranges of accessible states for tethered chains of lengths , 16, 32, and 64. The shaded areas approximate the ranges, and the symbols represent the realized states with the highest number of bead contacts for a given number of surface contacts . The results for were obtained in exact enumeration.
Density of states for tethered chains of length (top) and (bottom). The surfaces (small symbols connected by straight line segments) represent the log-density of states values, , as a function of surface contacts, , and bead-bead contacts, .
Most probable states for a given number of surface contacts and field . The lines represent the average number of bead contacts as function of the number of surface contacts for five fields , as indicated in the figure. The shaded areas surrounding the lines indicate the states that have significant probability of occupation; the sum of the probabilities associated with these states is . The symbols at the upper boundary of the graph indicate the maximum values of for given .
Average number of surface contacts per monomer and surface-contact fluctuations , as a function of the surface field for good-solvent conditions . The lines represent results from the evaluation of the density of states for chains of length (solid), 32 (dashed), and 16 (dash dotted), respectively. The graphs for are monotonously increasing; those for have a maximum in the transition region.
Ratio of perpendicular and parallel contributions to the square radius of gyration as a function of the surface field for good-solvent conditions . The lines represent results from the evaluation of the density of states and production data for chains of length (solid), 32 (dashed), and 16 (dash dotted); the inset shows an enlargement of the region where the lines cross. For clarity, error bars for our calculated values are shown only in the inset. (The error bars generally increase with increasing and decreasing ; for they are about twice as large as for the -range of the inset). The filled symbols with error bars represent Metropolis Monte Carlo results from this work. The open symbols represent Metropolis Monte Carlo results for chains of length , 40, and 80 by Descas et al. (Ref. 29).
Heat capacities per monomer, , as a function of the reduced temperature . The solid , short-dashed , and dash-dotted lines represent results for chains tethered to a hard surface . The long-dashed and dotted lines represent results for free chains (Refs. 32 and 38). The estimated uncertainties of the results for the tethered chains are smaller than the line thickness, except for at very low temperatures, , where they correspond to about twice the line thickness.
Monomer-monomer contact fluctuations, , as a function of the field for a hard surface, . The lines represent results from the evaluation of the density of states for tethered chains of length (solid), 32 (dashed), and 16 (dash dotted).
Monomer-monomer contact fluctuations, , as a function of the field for a very attractive surface, . The lines represent results from the evaluation of the density of states for tethered chains of length (solid), 32 (dashed), and 16 (dash dotted).
Chain collapse for large surface field. The top panel shows the average square bond lengths , radius of gyration divided by the chain length , and the average number of monomer-monomer contacts per bead as a function of the monomer-contact field for a chain of length and a surface field . The bottom panel shows how these quantities change as the chain undergoes the collapse transition. The dashed and dash-dotted lines represent absolute values of the numerical derivatives and , respectively. The solid line represents as in Fig. 9.
Examples for compact conformations of chains of length . (a) A highly ordered three-dimensional conformation representative of the desorbed compact (DC) region of the phase portrait in Fig. 12. (b) A highly ordered two-dimensional (single-layer) conformation representative of the adsorbed compact (AC) region of the phase portrait. In both diagrams, the size of the circles corresponds to the hard core diameter of the beads; the bonds are shown as wide lines.
Phase portrait for a tethered chain of length in the space of the surface field and bead-contact field . The regions are named as in Fig. 1. The solid lines represent maxima of surface and bead-contact fluctuations, as explained in the text. The dashed lines are an estimate for the location of the coil-globule transition from the “shoulder” on the susceptibility . The dotted lines represent shallow maxima in the susceptibility that depend sensitively on the details of the available compact chain conformations. In the shaded area near the center of the diagram, the susceptibility “landscapes” are too complex to identify all of the maxima clearly. This is why some of the lines end rather than merge with other lines.
Umbrella sampling parameters and some density-of-states characteristics for chains of length , , and . is the number of results for which umbrella sampling simulations were performed, is the length of the simulations in Monte Carlo steps, is the number of states sampled, and and represent the largest number of bead-bead contacts for and surface contacts, respectively. The values for the range of the log-density of states , and its uncertainties represent results after umbrella sampling. For and , the states in this table are believed to be the complete set; for , only states included in our evaluation are represented.
Production parameters for chains of length , , and . The table entries represent the simulation length in MC steps, where configurations are evaluated every ten MC steps. The left column indicates the type of simulation. For simulations sampling with the density of states, the number of replicas is indicated and, in one case, the maximum number of bead contacts considered. For simulations with the Metropolis acceptance criterion, the values for the field variables and are shown.
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