3.2 Mechanically Interlocked Calix[4]arene Dimers
Under External Force
Thermal fluctuations in macroscopic systems close to a local equilibrium are
merely a source of noise that can usually be neglected in everyday physics. In
contrast to macroscopic systems, small systems are largely influenced by thermal
fluctuations. At macroscopic scales, the experimental outcomes are essentially
independent of the repetition of thermodynamic manipulations following the identical protocol, but the situation is different at microscopic scales, where outcomes
from repetitions of an identical experimental protocol vary substantially
[133]. Recent theoretical developments in nonequilibrium physics have shown
how, by using these fluctuations, it is possible to recover free energy differences
and energy landscapes from experiments carried out far from equilibrium [104,
134–137]. The methods usually require large sampling of rather rare events, which
poses a great challenge to experimentalists. Experiments suitable for verifying
modern theories of nonequilibrium statistical mechanics like the fluctuation theorem and the Jarzynski relation need to fulfill two fundamental requirements
[138]. On the one hand, manipulative devices such as optical/magnetic tweezers
or force microscopes with a bendable cantilever are needed that allow fixing a
single state variable such as a defined force or distance, while other variables are
allowed to fluctuate [139]. Moreover, since sampling of rare events is necessary to
obtain the free energy from out-of-equilibrium experiments, a large number of
repetitions are required to reconstruct the underlying potential. Therefore, stable
experimental set-ups and special molecules with defined states are needed to permit
a large number of realizations with variable outcome.
Therefore, tunable model systems of sufficient smallness are required that can
be subjected to defined external mechanical perturbations, giving access to both
the equilibrium and nonequilibrium regimes. This allows verification of the free
energy differences computed from nonequilibrium conditions by switching the
system into quasi-equilibrium conditions, providing the free energy differences
directly.
Although supramolecular assemblies are ideally suited to study the physics of
small systems under external load, only a few are appropriate for the study of
reversible and irreversible transformations. Most examples are from biomolecules
such as proteins and nucleic acids [93, 140]. Along these lines, Bustamante and
coworkers were the first to establish a reversible model system to apply Jarzynki’s
relation to compute the free energy difference from out-of-equilibrium
experiments [141].
Recently, a supramolecular model system was successfully established that allows
the assessment of different regimes of externally stimulated stochastic barrier crossing, ranging from quasi-equilibrium to nonequilibrium bond breakage. This has been
achieved by creating a modular molecule that prevents irreversible bond rupture by
mechanically limiting the separation distance using entangled loops (Fig. 30)
[95]. The mechanical lock limits the distance of the two binding partners, raising
Mechanical Properties of Single Molecules and Polymer Aggregates
39
Under External Force
Thermal fluctuations in macroscopic systems close to a local equilibrium are
merely a source of noise that can usually be neglected in everyday physics. In
contrast to macroscopic systems, small systems are largely influenced by thermal
fluctuations. At macroscopic scales, the experimental outcomes are essentially
independent of the repetition of thermodynamic manipulations following the identical protocol, but the situation is different at microscopic scales, where outcomes
from repetitions of an identical experimental protocol vary substantially
[133]. Recent theoretical developments in nonequilibrium physics have shown
how, by using these fluctuations, it is possible to recover free energy differences
and energy landscapes from experiments carried out far from equilibrium [104,
134–137]. The methods usually require large sampling of rather rare events, which
poses a great challenge to experimentalists. Experiments suitable for verifying
modern theories of nonequilibrium statistical mechanics like the fluctuation theorem and the Jarzynski relation need to fulfill two fundamental requirements
[138]. On the one hand, manipulative devices such as optical/magnetic tweezers
or force microscopes with a bendable cantilever are needed that allow fixing a
single state variable such as a defined force or distance, while other variables are
allowed to fluctuate [139]. Moreover, since sampling of rare events is necessary to
obtain the free energy from out-of-equilibrium experiments, a large number of
repetitions are required to reconstruct the underlying potential. Therefore, stable
experimental set-ups and special molecules with defined states are needed to permit
a large number of realizations with variable outcome.
Therefore, tunable model systems of sufficient smallness are required that can
be subjected to defined external mechanical perturbations, giving access to both
the equilibrium and nonequilibrium regimes. This allows verification of the free
energy differences computed from nonequilibrium conditions by switching the
system into quasi-equilibrium conditions, providing the free energy differences
directly.
Although supramolecular assemblies are ideally suited to study the physics of
small systems under external load, only a few are appropriate for the study of
reversible and irreversible transformations. Most examples are from biomolecules
such as proteins and nucleic acids [93, 140]. Along these lines, Bustamante and
coworkers were the first to establish a reversible model system to apply Jarzynki’s
relation to compute the free energy difference from out-of-equilibrium
experiments [141].
Recently, a supramolecular model system was successfully established that allows
the assessment of different regimes of externally stimulated stochastic barrier crossing, ranging from quasi-equilibrium to nonequilibrium bond breakage. This has been
achieved by creating a modular molecule that prevents irreversible bond rupture by
mechanically limiting the separation distance using entangled loops (Fig. 30)
[95]. The mechanical lock limits the distance of the two binding partners, raising
Mechanical Properties of Single Molecules and Polymer Aggregates
39