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# Molecular dynamics of a grid-mounted molecular dipolar rotor in a rotating electric field

Contributed by Josef Michl

## Abstract

Classical molecular dynamics is applied to the rotation of a
dipolar molecular rotor mounted on a square grid and driven by rotating
electric field *E*(ν) at *T* ≃ 150
K. The rotor is a complex of Re with two substituted
*o*-phenanthrolines, one positively and one negatively
charged, attached to an axial position of Rh in a
[2]staffanedicarboxylate grid through
2-(3-cyanobicyclo[1.1.1]pent-1-yl)malonic dialdehyde. Four regimes
are characterized by *a*, the average lag per turn:
(*i*) synchronous (*a* <
1/*e*) at *E*(ν) =
|*E*(ν)| > *E*_{c}(ν)
[*E*_{c}(ν) is the critical field strength],
(*ii*) asynchronous (1/*e* <
*a* < 1) at *E*_{c}(ν)
> *E*(ν) >
*E*_{bo}(ν) >
*kT*/μ, [*E*_{bo}(ν)
is the break-off field strength], (*iii*) random driven
(*a* ≃ 1) at
*E*_{bo}(ν) > *E*(ν)
> *kT*/μ, and (*iv*) random
thermal (*a* ≃ 1) at
*kT*/μ > *E*(ν). A fifth
regime, (v) strongly hindered, *W* >
*kT*, *Eμ*, (*W* is the
rotational barrier), has not been examined. We find
*E*_{bo}(ν)/kVcm^{−1} ≃
(*kT*/μ)/kVcm^{−1} +
0.13(ν/GHz)^{1.9} and
*E*_{c}(ν)/kVcm^{−1} ≃
(2.3*kT*/μ)/kVcm^{−1} +
0.87(ν/GHz)^{1.6}. For ν > 40 GHz, the rotor
behaves as a macroscopic body with a friction constant proportional to
frequency, η/eVps ≃ 1.14 ν/THz, and for ν < 20
GHz, it exhibits a uniquely molecular behavior.

Molecular construction kits promise access to giant molecules shaped like regular grids and scaffolds carrying active and/or mobile groups (1–4). A sturdy artificial square grid polymer, albeit irregular and only ∼150 nm across, has been reported (5, 6), and time appears right to use computer simulation of these “designer solids” to analyze the role of thermal motion in molecular machinery and to identify the best synthetic targets.

One of the active elements proposed (4, 7) for incorporation in these molecular scaffolds is a dipolar rotor, capable of rotational motion in response to an outside driving force. Molecular rotors driven unidirectionally by light (8), or bidirectionally by heat (9–12) or chemically (13, 14) have been synthesized. Alternating electric field has been used to affect intramolecular motion (15), and intense laser field of rapidly rotating linear polarization has been used to drive the rotation of a chlorine molecule (16). Free hydroxyl groups on oxide surfaces behave as two-dimensional arrays of interacting rotors, as do phospholipids in certain membranes and dipolar molecules adsorbed on flat surfaces or intercalated in layered solids (17). Biological motors (18) are under study. The velocity at which a dissolved chiral molecule is propelled through a solution by circularly polarized microwaves has been theoretically estimated (19–21).

Previously (22), we examined computationally the unidirectional motion of a grid-mounted propeller-shaped rotor driven by a stream of gas. This windmill rotor acted as a microphone, transducing linear motion of gas molecules into rotational motion of electric charges. A propeller-shaped rotor driven by a rotating electric field in the presence of a gas might produce a pressure differential, acting as a loudspeaker or a gas pump.

The dielectric response of a dipolar rotor is intrinsically nonlinear and therefore theoretically and possibly even practically interesting. Analyses (23–25) of a one-dimensional string of rotors interacting by electrostatic forces suggest that solitary waves may be possible. Such analyses benefit from a phenomenological characterization of the rotor, e.g., by its friction constant. The statistical physics of interacting rotors in two dimensions is likely to be of special importance (17), considering the strong current interest in two-dimensional spin glasses (26).

We examine the motion of a single dipolar rotor mounted on a square
grid and driven in vacuum by a rotating electric field. Two prior
studies (27, 28) simulated the motion of a dipolar rotor consisting of
two concentric carbon nanotubes in a linearly oscillating electric
field. Our work differs in several respects: (*i*) the
synthesis of our rotor is feasible by procedures known today,
(*ii*) the propeller is mounted on a molecular grid,
(*iii*) we use a rotating electric field, avoiding the
irregular changes in the sense of rotation that dominated the earlier
results, (*iv*) we apply much weaker fields, and
(*v*) we fit a model to the results and extract a value of an
effective friction constant.

We use the smaller of the rotors and the finer of the grids described
previously (22) (Fig. 1). The dipole
moment of the rotor μ = |**μ**| is ∼42 debye, its
moment of inertia *I* is ∼1.5 ×
10^{4} *u*Å^{2}
(*u* is the atomic mass unit), and its 4-fold rotational
barrier on the grid is *W* = 0.3 ± 0.2 kcal/mol.
We vary the frequency ν (angular frequency ω = 2πν,
ω = |**ω**|, with **ω** directed in the
positive sense of the rotation axis *z*) and the strength of
the rotating electric field *E*(ν) =
|** E**(ν)|. Exploratory runs at various temperatures

*T*showed that at 150 K the rotor behaves well, and further simulations were initiated at this temperature.

## Method of Calculation

The classical molecular dynamics computer program uses the UFF
potential energy and charge equilibration scheme of Rappé
*et al.* (29, 30), which ignores polarization and treats all
elements in the periodic table at the cost of reduced accuracy. It does
not correct for the negligible radiative losses due to moving charges,
nor for quantum mechanical effects, which are of unknown magnitude but
are not expected to change the results qualitatively. We did not try to
compensate for the latter by freezing high-frequency motions of C—H
bonds, which probably causes friction to be overestimated, whereas the
finite size of the modeled grid causes it to be underestimated.

The square grid fragment modeled (Fig. 1) consisted of 12 rods,
[2]staffanes terminally substituted with two carboxylates (4),
attached to nine dirhodium tetracation (31, 32) connectors whose axial
positions were occupied by CH_{3}CN ligands on top
and HCN ligands at the bottom, except for the central bottom position
used to hold the nitrile group at the end of the propeller axis. The
rotor has been described in ref. 22. The axial attachment of the
nitrile group to rhodium is weak but adequate for the length of the
simulation runs used. No dissociation events were observed when a Morse
potential was used for the Rh—N bond. In practice, the structures
used will need to be stable indefinitely at the operating temperature.

The computational procedure (22) was simplified by the absence of gas
atoms. Constant energy molecular dynamics with external electric field
was used (no velocity scaling). Before simulation, the molecular
structures were well optimized by using truncated Newton optimization.
As usual, initial velocities were assigned according to a
Maxwell-Boltzmann velocity distribution. Different starting geometries
and velocity distributions were used in some simulations to sample more
space. The actual initial temperatures therefore differed slightly from
150 K and from one run to another. Integration time steps of 1.1, 1.8,
and 2.8 fs were tested and gave the same results; thereafter, the step
was fixed at 2.1 fs. A typical total run time
*t _{tot}* was sufficient for 5–350 full
revolutions of the electric field vector

**(0.1–3.5 ns, most often 0.5–1 ns). The total simulated time was over 100 ns (171 runs).**

*E*The rotational barrier *W* was calculated as the energy
difference of the whole structure at a geometry optimized with the
angle between the rotor dipole and the grid rods constrained once at 0
and once at 45°. To obtain sufficient accuracy it was essential to
use a 50-Å cutoff for electrostatic interactions; in the molecular
dynamics runs the cutoff was 30 Å.

The effect of an external grid support was modeled by fixing the
position of the Rh atom to which the rotor axle is attached (the origin
of the coordinate system), and constraining the 12 bridgehead H atoms
at the grid periphery to the *xy* plane. The homogeneous
external electric field ** E** rotated around the

*z*axis and its components were

*E*=

_{x}*E*sin(ω

*t*+ π/4),

*E*=

_{y}*E*cos(ω

*t*+ π/4),

*E*= 0, where

_{z}*t*is time. The dipole was initially oriented along the

*y*axis. Time averages 〈

*q*〉 =

*t*∫

*q*(

*t*)

*dt*of various properties

*q*were calculated separately over each run of duration

*t*

_{tot}. The properties monitored were the temperature of the rotor

*T*

_{r}(

*t*) (overall rotation of the rotor was subtracted from the motion of the atoms,

*cf.*ref. 22) and of the grid

*T*

_{g}(

*t*), the

*z*components of the rotor angular momentum

*M*(

_{z}*t*) and moment of inertia

*I*(

_{z}*t*), the projection of the rotor dipole moment into the

*xy*plane

**μ**

_{xy}(

*t*), the number of turns

*m*the rotor was behind the field vector, and the cumulative angle α(

*t*) (−∞ < α < ∞) by which it lagged behind the driving electric field (measured between the projections of the dipole moment vector of the rotor

**μ**and of the rotating electric field vector

**into the**

*E**xy*plane, positive when the rotor lags behind the field).

The performance of the rotor was judged by: (*i*) The mean lag
per turn *a* =
α_{tot}/2π*n*, where
α_{tot} =
α(*t*_{tot}) is the final value of the
lag angle α after *n* turns of the field. (*ii*)
The time-averaged lag angle 〈α〉. (*iii*) The
time-averaged lag per turn *a*′ = 〈α〉/π*n*
(similar to *a*; not used in the discussion). (*iv*)
The average rotational frequency of the rotor
ν_{rot} =
〈*M _{z}*〉/(2π〈

*I*〉). (

_{z}*v*) The rms fluctuation of the lag angle 〈Δα〉 = (〈α

^{2}〉 − 〈α〉

^{2})

^{1/2}. (

*vi*) A measure of the average torque on the rotor, 〈sinα〉. (

*vii*) A measure of the average polarization of the rotor in the rotating coordinate system, 〈cosα〉.

## Results

Sample results are collected in Table
1. The average projection of the rotor
dipole moment into the *xy* plane is
**μ**_{xy} ≃ 41.6 debye, and the average
*z* component of the moment of inertia of the rotor is
1.5 × 10^{4}
*u*Å^{2}, occasionally reaching values up
to 4.5 × 10^{4}
*u*Å^{2}, due to centrifugal forces,
internal motion within the rotor, and its overall irregular pendular
motion. Whereas μ_{xy} grows only slightly with
increasing rotational frequency, 〈*M _{z}*〉
is more sensitive to the mode of overall motion and often increases
significantly with the rotational frequency. The
〈

*M*〉 values are mostly negative, i.e., the rotor is moving in the same sense as the field. In most cases, the temperatures

_{z}*T*

_{r}(

*t*) and

*T*

_{g}(

*t*) remained equal to the initial temperature within a few K. A significant increase was generally observed only at frequency above 100 GHz at fields stronger than 3,000 kV⋅cm

^{−1}. The highest observed temperature was 380 K. In longer simulations (∼2 ns), however, a slight temperature increase by up to 20 K was observed even for weaker fields. No provision has been made to remove the heat generated. The actual average temperature was evaluated for each simulation and taken into account during the evaluation of the data.

At eight frequencies from 3.2 to 200 GHz, *E* was varied from
100 to 7,000 kV/cm, and at six field strengths from 430 to 5208
kV⋅cm^{−1}, ν was varied from 2 to 400 GHz
(Table 1). Fig. 2 presents the rotor
motion in four representative runs, and Fig.
3 shows the variation of
*M _{z}*(

*t*) and α(

*t*) in two of them. The dependence of four measures of rotor performance on

*E*is similar at all frequencies. The results obtained at ν = 3 GHz are shown in Fig. 4. The estimated field strength above which

*a*begins to deviate detectably from unity is denoted

*E*

_{bo}, and the critical field strength at which

*a*drops below 1/

*e*is denoted

*E*

_{c}. Within the accuracy of this determination, these field strengths could just as well be read off the plot of 〈cosα〉 or ν

_{rot}/ν. The plot of 〈sinα〉 is too flat and less useful. There is an uncertainty in the definition of the break-off field

*E*

_{bo}unless the smallest deviation of

*a*from unity that can be reliably detected is specified. This is a function of the length of the molecular dynamics runs; the uncertainty that pertains to the present results (Table 2) is marked by bars in Fig. 5.

The plots of log *E*_{bo} and log
*E*_{c} against log ν (Fig. 5) are
approximately linear above ∼40 GHz;
*E*_{bo}(ν)/kV⋅cm^{−1}
≃ (*kT*/μ)/kV⋅cm^{−1} +
0.13(ν/GHz)^{1.9} and
*E*_{c}(ν)/kV⋅cm^{−1}
≃
(2.3*kT*/μ)/kV⋅cm^{−1}
+ 0.87(ν/GHz)^{1.6} (*k* is the
Boltzmann constant, *T* = 150 K). Below ∼40 GHz,
*E*_{bo} becomes independent of the
frequency, *E*_{bo} ≃
*kT*/μ = 140
kV⋅cm^{−1}, whereas
*E*_{c} converges more slowly to a higher
limit, *E*_{c}(0) ≃ 2.3
*kT*/μ. *E*_{c} is
roughly equal to 2*E*_{bo} (ν); at
the highest fields this ratio decreases.

In 35 additional runs we examined the field-free decay of the
rotational motion. After a 15-ps period of thermal equilibration,
additional velocity vectors corresponding to a rotational excitation at
a frequency at 155 GHz in either direction were instantaneously
assigned to all atoms of the rotor and the subsequent motion was
monitored. The *z* component of the angular momentum decayed
from its initial value *M _{z}*(0), rapidly at
first and then increasingly slowly, and ceased to be detectable after
200 ps. When the normalized decay of the angular momentum,

*M*/

_{z}*M*(0), averaged over the 35 runs (Fig. 6), was forced to fit a single exponential, it yielded a relaxation time τ

_{z}_{0}≃ 83 ps (decay to 1/

*e*of the initial value).

## Discussion

An ideal simulation of the rotor would run forever,
*t*_{tot} = ∞. In this limit, synchronous
rotation (*m* = 0) would be characterized by
*a* = 0 and ν_{rot} =ν, but we
accept values up to *a* = 1/*e* as
synchronous. The quantities 〈α〉, 〈δα〉, 〈sinα〉, and
〈cosα〉 would be physically meaningful measures of the average lag
angle, its fluctuation, the average torque, and the average alignment
of the dipole with the rotating field, respectively. Asynchronous
rotation (*m* ≠ 0) would be characterized by 1 >
*a* > 0 (1 > *a* >
1/*e*), and ν_{rot} <ν, where
*a* provides a measure of the probability with which the rotor
skips a turn of the field and raises the total lag angle
α_{tot} by 2π. If the probability that the rotor
breaks off in a single turn (and loses memory of this event before the
next turn) is *P*, *a* = *P* +
α/2π*n* ≃ *P*. In the thermal regime,
ν_{rot} would approach zero and *a* would
approach unity. Our simulations are not converged in the statistical
sense, causing a scatter of the results around the values expected for
*t*_{tot} = ∞ (Table 1).

### A Simple Model.

The results are compatible with intuitive expectations based on a simple classical model, which invokes friction and random thermal motion as the two factors opposed to synchronous rotation.

#### The limit of zero temperature and/or high field.

For negligible thermal motion (*T* → 0 and/or *E*
→ ∞), the lag angle α(*t*) is determined by friction. In
a stationary state, the value of α(*t*) =
α_{0} will be such that the rotor driving torque
** E** ×

**μ**compensates the drag torque, −

**×**

*E***μ**= η

**ω**, where the angular frequency

**ω**is viewed as a vector and η is the friction constant. Therefore, 1 If α

_{0}< π/2, the rotor does not break off at all (

*a*= 0). As

*E*is reduced or ν increased and the lag angle α

_{0}attains the critical value of π/2, the break-off point is reached, and friction prevents synchronous rotation (

*a*= 1). For a frequency independent η, the break-off frequency ω

_{0}would be proportional to field strength, ω

_{0}=

*Eμ*/η, and the zero-temperature field strength

*E*

_{0}, below which the rotor no longer follows the rotating field at zero temperature, would be proportional to frequency (at

*T*= 0,

*E*

_{bo}=

*E*

_{c}=

*E*

_{0}): 2 The approximately quadratic high-field limit frequency dependence of

*E*

_{bo}and

*E*

_{c}observed at 150 K therefore suggests that for our rotor η is proportional to frequency.

#### Finite temperature and field.

Above absolute zero and at driving electric fields that are not
excessively strong, the lag angle α fluctuates about its mean value
〈α〉. As *E* is reduced, the instantaneous value of α
reaches the critical value of π/2 at a temperature-dependent rate
even before 〈α〉 reaches π/2, and the rotor misses a turn.
Because at *T* > 0 there is always a nonzero probability
of skipping a turn, the break-off field
*E*_{bo} at which *a* begins to
deviate noticeably from unity is larger than
*E*_{0}, and the critical field
*E*_{c}, where it drops to
1/*e*, is even larger. The ratio
*E*_{c}/*E*_{bo}
is a measure of the abruptness of the transition from perfect rotor
behavior to irregular motion and should increase with temperature. It
seems to be largely independent of ν.

The rotor also may gain an extra turn during a favorable thermal
fluctuation. Its potential energy *U* is the sum of a friction
term *U*_{f} = −ηωα and a
polarization energy term *U*_{p} =
*Eμ*cosα. A plot of *U* against the lag angle in
a coordinate system rotating at angular velocity ω exhibits a minimum
on the potential energy surface at a value α_{0}
defined by Eq. 1, and a maximum at the lag angle π/2 +
α_{0}, and is periodic (Fig.
7). When a turn is missed the rotor lag
angle α moves from stationary position α_{0} to
a new stationary position α_{0} + 2π, initially
against the force exerted by the electric field, and when a turn is
gained, it moves from α_{0} to
α_{0} − 2π. The activation energies
Δ*U* and Δ*U*′ are
3
The rates *p* and *p*′ at which α moves from
α_{0} to the critical value
α_{0} + π/2 or α_{0} −
π/2, thus causing a turn to be missed or gained in unit time,
respectively, are assumed to be
4
The probability of skipping a turn per field revolution is
*a* ≃ *m*/*n*, where
*m* = (*p* − *p*′)*t* is
the number of missed turns and *n* = *tν* is
the total number of field turns. Assuming a dead time
*t*_{dead} = 1/ν for each hopping event
during which the rotor may not start skipping another turn, the total
dead time is *m*/ν and thus the corrected number of missed
turns is *m* = (*p* −
*p*′)(*t* −
*t*_{dead}) = (*p* −
*p*′)(*t* − *m*/ν), which leads to
*m* = (*p* −
*p*′)*tν*/[ν + (*p* −
*p*′)] and
5
Eq. 5 does not yield a simple analytical formula for
the critical field strength *E*_{c}, and
*E*_{bo} has not been uniquely defined
mathematically. In the limit *T* → 0, Eq. 5
reverts to Eq. 2. In the limit *E* →
*E*_{0} and 2πηω ≫ *kT, a*
→ *A*/(*A* + ν), which causes a discontinuity
with the result *a* = 1 valid for *E* ≤
*E*_{0}. This is not significant if
*A* ≫ν, but the model cannot be used when
*A* ≃ ν.

#### The friction coefficient.

The effective friction coefficient η of the rotor was derived in four
ways, denoted η_{1}–η_{4}.

The *a*(*E*,ν,*T*) values from all 171
molecular dynamics simulations (Table 1, *cf.* points in Fig.
4) were fitted to Eqs. 3–5 using an adaptive
nonlinear least-squares algorithm (33), yielding
η_{1}. Because we could not fit the data with a
single friction constant independent of ν, we divided the frequency
range into 15 intervals with an independent value of
η_{1} in each. The *a*(*E*,ν)
values from the simulations (Fig. 8) and
the frequency dependence of *E*_{bo}(ν)
and *E*_{c}(ν) were fitted very well. The
resulting values of η_{1}(ν) (Table 2) are
approximately proportional to ν (Fig.
9), as anticipated from the discussion of
the high-field limit, η_{1}/eVps ≃ 1.1
ν/THz. Below ∼50 GHz friction is much smaller and is better
described by η_{1}/eVps ≃ 0.2 ν/THz.
Apparently, the coupling between the rotor rotational motion and
vibrational modes of the grid increases at frequencies above ∼50 GHz.

When exact proportionality between η_{1} and ν
was assumed and the proportionality coefficient was treated as the only
adjustable parameter in the least-squares fitting of the 171
*a*(*E*,ν) values, the result was
η_{1L}/eVps ≃ 1.14 ν/THz (Fig. 9).
The fit was now worse but still respectable, and the fit of the
*E*_{c}(ν) curve was still excellent
(Fig. 5). With η_{1L} proportional to ν, the
frequency dependence of *E*_{0}, the
zero-temperature limit of the break-off field calculated from Eq.
2, becomes linear in the log-log plot of Fig. 5, with a
slope of two. In high-field limit the finite temperature break-off
field strength curve *E*_{bo}(ν) fits it
asymptotically very nicely. In the low-field limit,
*E*_{bo}(ν) becomes frequency-independent
as it approaches *kT*/μ.

Above 40 GHz (where *E*_{0}μ ≃
*kT*), friction dominates and
*E*_{bo} can be identified with the value
*E*_{0} in Eq. 2, yielding
η_{2}(ν), in fair agreement with
η_{1}(ν) (Fig. 9).

To obtain η_{3}, the sine of the lag angle is
identified with the mean value 〈sinα〉 obtained in the simulation.
According to Eq. 1,
2πη_{3}/μ is equal to the slope of
the plot of 〈sinα〉 against ν/*E* when α is
dictated by friction and not by thermal fluctuations. We selected 58
runs in which the rotor followed the field most of the time
(〈sinα〉 > 0.0, 0.0 < *a* < 0.23) and
calculated η_{3} for 15 frequencies (Fig. 9). The
plot of 〈sinα〉 shows considerable scatter (Fig. 4), and the
resulting η_{3} is very inaccurate. Unlike
η_{1}, η_{1L}, and
η_{2}, it is almost frequency independent. It has
a low average value of 0.024 eVps, perhaps because the effective
friction constant for these selected runs differs from the rest. We do
not consider η_{3} reliable.

The rate of decay of 〈*M*_{z}〉 in the
absence of the driving field was used to estimate
η_{4} from τ_{0} =
*I*/η_{4}. A fit to a single
exponential (Fig. 6) yields η_{4} = 0.019 eVps in
the region of low frequencies examined, very close to
η_{1} and η_{3} (Fig. 9; in
this region η_{2} is not available). The decay
curve expected from η_{1} indeed agrees quite well
with that obtained from the molecular dynamics runs (Fig. 6). The much
faster decay observed for η_{1L} reflects the
overestimation of its value relative to η_{1} in
the frequency region below 50 GHz (Fig. 9).

#### The cause of friction.

All of the determinations of η suffer from statistical uncertainties
in the trajectory calculations. The friction constant
η_{1} is probably the most realistic, because even
in the synchronous motion limit the rotor fluctuates around the
direction of the field. The friction constant may depend on field
strength and temperature, but we were unable to prove such dependence.

The friction is attributed to a kinematic and electric-field coupling
of the rotor rotational states to the vibrational and rotational state
manifold of the molecular grid. As ν increases, it enters the region
of an increasing density of grid states and the coupling to the grid
becomes stronger, causing higher friction. It is possible that the plot
of η against ν contains specific resonances with particularly
strongly coupled low-frequency bending modes of the grid, but our
calculations are not accurate enough to reveal them unambiguously. The
fact that η_{1}(ν) is very small at low
frequencies suggests that at lower temperatures it may be possible to
rotate the rotor synchronously at even lower frequencies using
weaker fields, as long as the rotational barrier is low.

### Langevin Dynamics.

Numerically integrated Langevin dynamics including a stochastic force
(34) would provide a more accurate model. The Langevin equations of
Brownian motion are identical with the equations (35) describing the
current-voltage characteristics of a Josephson junction (JJ), with the
the JJ current variable *i* defined by *i* =
ω/ω_{0}. The JJ voltage *V* relative to
the current *i* can be expressed as
*V*/*i* =
〈dα/d*t*〉/ω ≈
α_{tot}/(*t*_{tot}ω)
= *a*. For our rotor the usual McCumber parameter
β_{C} and thermal parameter γ are
β_{C} =
*E*μ*I*η^{−2} and γ =
2π*kT*(*E*μ)^{−1}. Many
numerical results for JJ plots have been published, but we found little
information for the parameter values relevant for our rotor, 14 <
β_{C} < 140 and 0.17 < γ < 2.
Numerical solutions (36) of the Langevin equation for JJ circuits with
β_{C} = 1 and 4, and γ = 10, and
experimental observations (37) in a similar range yielded
*i* − *V* plots (figure 1 in ref. 36 and figure
11 in ref. 37) in qualitative agreement with our *a* −
ν plots, which correspond to *V*/*i* −
*i*. For β_{C} ≃ 4, the JJ results
predict that the motion of our rotor will be hysteretic: once regular
rotation is induced by an above-critical electric field, the field will
have to be reduced to about 70% of the critical value before the rotor
fails.

### Synchronous, Asynchronous, Random Driven, Random Thermal, and Hindered Regimes of Rotation.

We anticipate five regimes of molecular rotor behavior dictated by the
relative importance of random thermal forces described by
*kT*, of the strength (μ*E*) of maximum rotor-field
interaction, the friction constant η that describes the break-off
drag torque, and the rotational barrier height *W*. Presently,
we have investigated four of them.

#### Synchronous rotor regime.

When μ*E* >
μ*E*_{c},*kT*,*W*, the
rotor follows the rotating field slavishly and rotates at its frequency
ν (points above the *E*_{c} line in Fig.
5).

#### Asynchronous rotor regime.

When μ*E*_{c} > μ*E* >
μ*E*_{bo},*kT*,*W*, the
rotor turns at a frequency lower than ν because it occasionally skips
a turn (points between the *E*_{c} and
*E*_{bo} curves in Fig. 5). The decrease of
the ratio of the averaged rotational frequency of the rotor to that of
the field is abrupt (Fig. 4). Even for *a* values close to
unity the rotor still rotates significantly.

#### Random driven rotor regime.

When
μ*E*_{c},μ*E*_{bo}
> μ*E* > *kT*,*W,* friction is
excessive. The rotor cannot keep up with the rotating field and
performs irregular motion (points below the
*E*_{bo} curve and above the
*kT*/μ line in Fig. 5).

#### Random thermal rotor regime.

When *kT* > μ*E*,*W*, the rotor
exhibits nearly random thermal fluctuations with a slight preference
for rotation in the sense of the rotating field (points below the
*kT*/μ line in Fig. 5; at *T =* 150 K,
μ*E* equals *kT* at *E* = 140
kV⋅cm^{−1}). The independence of
*E*_{c} and
*E*_{bo} of ν in the limit of low
frequencies is unlike anything observed for macroscopic rotors at
ordinary temperatures. Molecular dynamics investigation of this regime
is unlikely to offer significant advantages over a purely statistical
description (19) (a freely rotating dipolar solute in a solvent of
relaxation constant Γ has an average angular velocity (ω/2)[1 +
(ω/Γ)^{2}]^{−1}
(μ*E*/*kT*)^{2} to second
order in μ*E*/*kT*.

#### Hindered rotor regime.

When *W* > *kT*,μ*E*, the rotor will
jump infrequently from one to another minimum on the potential energy
surface. At 150 K, *kT* = 0.3 kcal/mol, which is
comparable with our estimate of the barrier height *W*. As a
result, for *E* < ∼140
kV⋅cm^{−1}, the effect of the barrier, and
possibly also quantization effects, will have to be included in the
modeling. Our simulations have been performed at *kT* ≥
*W* or at a higher field strength, and we do not address this
issue.

## Conclusions

Molecular dynamics simulations and simple modeling provide a detailed view of the interplay of the driving force, friction, and thermal motion acting on a dipolar molecular rotor, likely to be informative for “single-molecule machines” in general. The regimes examined at 150 K range from macroscopic (synchronous or random driven rotor) to molecular (random thermal rotor). The derived phenomenological friction constant should facilitate studies of dipolar rotor arrays.

## Acknowledgments

We are grateful to Prof. John C. Price for a helpful discussion. This project was supported by the National Science Foundation (CHE-9871917), the European Office of Aerospace Research and Development (SPC-98-4031), the J. Heyrovský Institute, and the Grant Agency of the Academy of Sciences of the Czech Republic (B4040006/00).

## Abbreviation

- JJ,
- Josephson junction

- Accepted February 28, 2001.

- Copyright © 2001, The National Academy of Sciences

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