10.2. Purely mechanical machines[[Student version, January 17, 2003]] 369
b. Show that athighforce (but still much smaller than/L), Equation 10.7 reduces to
(fL
kBT) (^2) D
L
e−fL/kBT. (10.8)
The last result establishes Sullivan’s fourth claim (forward stepping rate contains an exponential
activation-energy factor), in the perfect-ratchet limit (backward stepping rate equals zero).
Though we only studied the perfect-ratchet limit, we can now guess what will happen more
generally. Consider the equilibrium case, wheref=/L.Atthis point the activation barriers to
forward and reverse motion are equal. Your result in Your Turn 10d(b) suggests that then the
forward and reverse rates cancel, giving no net progress. This argument should sound familiar—it
is just the kinetic interpretation of equilibrium (see Section 6.6.2 on page 194). At still greater
force,f>/L,the barrier to backward motion is actually smaller than the one for forward motion
(see Figure 10.11d), and the machine makes net progress to the left. That was Sullivan’s second
claim.
Summary The S-ratchet makes progress to the right whenf/L,then slows and reverses as
weraise the load force beyondf=/L.
The S-ratchet may seem rather artificial, but it illustrated some useful principles applicable to
any molecular-scale machine:
1.Molecular-scale machines move by random-walking over their energy landscape, not by deter-
ministic sliding.
2.They can pass through potential-energy barriers, with an average waiting time given by an
exponential factor.
3.They can store potential energy (this is in part what creates the landscape), but not kinetic
energy (since viscous dissipation is strong in the nanoworld, see Chapter 5).
Point (3) above stands in contrast to familiar macroscopic machines like a pendulum clock, whose
rate is controlled by the inertia of the pendulum. Inertia is immaterial in the highly damped
nanoworld; instead the speed of a molecular motor is controlled by activation barriers.
Our study of ratchets has also yielded some more specific results:
a. A thermal machine can convert stored internal energyinto directed motion,
if it is constructed asymmetrically.
b. But structural asymmetry alone is not enough: A thermal machine won’t
go anywhere if it’s in equilibrium (periodic potential, Figure 10.11a). To get
useful work we must push it out of equilibrium, by arranging for a descending
free energy landscape.
c. A ratchet’s speed does not increase without bound as we increase the drive
energy.Instead, the speed of the unloaded ratchet saturates at some limiting
value (Equation 10.7).
(10.9)
Youshowed in Your Turn 10d that with a load, the limiting speed gets reduced by an exponential
factor relative to the unloaded 2D/L. This result should remind you of the Arrhenius rate law
(Section 3.2.4 on page 79). Chapter 3 gave a rather simpleminded approach to this law, imagining
asingle thermal kick carrying us all the way over a barrier. In the presence of viscous friction such