**Nanomedicine,
Volume I: Basic Capabilities**

**©
1999 Robert A. Freitas Jr. All Rights
Reserved.**

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

**4.4.3 Single-Proton
Massometer**

A nanopendulum may also be used to measure molecular-scale
masses. Torsion, helical spring, and vibrating rod mechanisms all will work,
but one of the most efficient designs for weighing the smallest masses may be
the coiled suspension-spring cantilever massometer illustrated schematically
in Figure
4.6, which is analogous in principle to the traditional quartz crystal microbalance.^{2896}
In this design, a beam of length L_{beam} and circular cross-sectional
radius R_{beam} is hinged at one end and carries a sample holder of
solid volume V_{sample} at the other end. The beam is supported (a variable
distance L_{spring} from the hinge) by a spring consisting of a movable
suspension rod of total length l_{rod}, circular cross-sectional radius
r_{rod}, and spring constant k_{s} ~ p^{2}
r_{rod}^{4} E G / (G l_{rod}^{3} + E p
r_{rod}^{2} l_{rod}) using spring (rod) material of
density r, Young's modulus E, and shear modulus G.*
The natural resonance period for this unloaded system is T_{0} = (2
p L_{beam}/L_{spring}) (m_{0}/k_{s})^{1/2}
= Z m_{0}^{1/2}, where Z = (2 L_{beam} / L_{spring}
r_{rod}^{2}) ((G l_{rod}^{3} + E p
r_{rod}^{2} l_{rod}) / E G)^{1/2} and m_{0}
= r (V_{sample} + p
L_{beam} R_{beam}^{2} + p
l_{rod} r_{rod}^{2}). Gravitational loading is negligible
for suspension system mass m << (k_{s} L_{spring}^{2}
/ g L_{beam}^{2}) ~ 10^{-15} kg; m ~ 10^{-20}
kg in the exemplar system.

* The l_{rod}^{-3} dependence
of k_{s} is not strictly valid for a folded structure because the forces
at the end of a folded spring have a smaller mechanical advantage, so the structure
acts somewhat stiffer. Further investigation is warranted. However, a large
number of alternative low-stiffness structures are available including van der
Waals contacts with low loading, gas springs, diamagnetic traps and actively
controlled electrostatic traps [J. Soreff, personal communication, 1997].

After adding a test mass Dm to
the sample holder, the resonance period increases to T_{m} = Z (m_{0}
+ Dm)^{1/2}, which may be detected if T_{m}
- T_{0} __>__ Dt_{min} ~ 1
nanosec, the minimum convenient time interval (Section 10.1).
Hence the minimum detectable molecular weight DMW,
expressed in proton masses (m_{p} = 1.67 x 10^{-27} kg), is

In Figure
4.6, beam and rod are coiled in two dimensions (although three-dimensional
coiling, more difficult to depict, provides maximum compactness). L_{beam}
and l_{rod} are chosen to minimize total coil area A_{coil}
= 2 R_{beam} L_{beam} + 2 r_{rod} l_{rod}. Using
aluminum for the rod and beam (E = 6.9 x 10^{10} N/m^{2}, G
= 2.6 x 10^{10} N/m^{2}, r = 2700
kg/m^{3}), L_{beam} = 1200 nm, l_{rod} = 1920 nm, r_{rod}
= 0.5 nm, V_{sample} = 1 nm^{3}, and Dt_{min}
= 10^{-9} sec, then DMW ~ 1 proton for L_{spring}
= 1 nm >> Dx_{min} = 0.01 nm, the minimum
measurable displacement (Section 4.3.1). (For L_{spring}
> 1 nm, physiological ~1 g accelerative loads on a ~1 micron nanorobot in
which the sensor is embedded produce undetectable accelerative beam displacements
< Dx_{min}.)

Moving the suspension rod closer to the sample holder reduces
device sensitivity. For L_{spring} = L_{beam}, minimum DMW
~ 1200 protons, permitting moderately large molecules to be weighed in minimum
time (~10^{-6} sec). Using a very thin sensor configuration (h_{sensor}~20
nm) and coils occupying ~50% of sensor cavity space, sensor dimensions are (79
nm)^{2} x 20 nm giving a massometer volume ~10^{5} nm^{3}
or ~0.01% of the total volume of a 1 micron^{3} nanorobot. Sensor mass
is ~ 3 x 10^{-19} kg, including diamondoid housing and prorated share
of support mechanisms. If one measurement cycle including sample loading/unloading
operations can be completed every ~10 microsec, then the device processes its
own mass of glucose molecules in ~10 sec. Note that biological molecules weighed
in air or in vacuo may lose water of hydration, whereas weighing in water presents
other complications that merit further investigation.

J. Soreff points out that for a molecule on a spring, the precision of the measurement is linked to the sharpness of the resonance, which is optimized to the degree the vibrational frequency of the spring/molecular mass system is kept well below the vibrational resonance frequencies of the molecule itself. The following discussion reports the results of a simple lumped-element analysis of the effects of internal vibration on the massometer completed for the author by J. Soreff, which confirms the previously described performance estimates for the massometer.

Suppose that it is desired to weigh some mass M to a precision
DM by attaching it to a weak measuring spring (spring
constant k_{weak}) and measuring the shift in resonant frequency. The
shift in frequency due to coupling to an internal mode of the sample must be
less than (1/2) (DM / M) (1 / 2 p)
(k_{weak}/M)^{1/2}. If the sample has some lowest internal mode
frequency n_{int}, then approximating the
sample as two M/2 masses separated by x_{strong} connected by a strong
spring (k_{strong}) gives n_{int}
= (1/p) (k_{strong}/M)^{1/2}. If
this sample is attached to a weak measuring spring of length x_{weak},
which is in turn attached to a rigid support, then the equations of motion are:

_{}
{Eqn. 4.26}

Setting x_{weak} = e^{st} and x_{strong}
= R e^{st} where t = time, s = time constant, and R = -(1 + (2 k_{weak}
/ M s^{2})), and since the angular frequencies of the two springs are
w_{strong} = 2 p n_{int}
= 2 (k_{strong}/M)^{1/2} and w_{weak}
= (k_{weak}/M)^{1/2}, then the lower frequency solution for
w_{weak} << w_{strong}
may be approximated by s = i w_{weak} (1
- w_{weak}^{2} / 2 w_{strong}^{2})
and so the (undesired) fractional change in frequency due to the flexibility
of the object to be weighed is

For a molecule sample size of L_{sample}, n_{int}
~ v_{sound} / L_{sample} ~ 500 GHz for v_{sound} ~ 1500
m/sec and L_{sample} ~ 3 nm; M = r L_{sample}^{3}
= 2.7 x 10^{-23} kg for r = 1000 kg/m^{3}.
To detect the mass difference of a single proton in a sample molecule, sensor
Q = M / m_{p} = 16,000; from Eqn. 4.28,
the sensor may operate at a factor of Q^{1/2} below n_{int},
or n_{spring} ~ n_{int}
/ Q^{1/2} ~ 4 GHz. A period of 0.25 nanosec is slightly too short to
allow a phase shift of Dt_{min} ~ 1 nanosec,
since the period is only one-quarter of this. However, a phase shift of p
radians should accumulate in t_{meas} = Q / 2 n_{spring}
~ 2 microsec and should be detectable by looking at magnitudes of forces. Interestingly,
an SPM microprobe of circular cross-section and exponential profile with a 0.1
nm^{2} tip has been proposed that could also reach ~proton mass detection
sensitivity at room temperature and 1 atm pressure with an eigenmode frequency
of 10 GHz;^{1195} an electrically-driven
carbon nanotube-based resonant-beam balance demonstrated ~10^{-17} kg
(~10^{10} proton or ~0.01 micron^{3} mass) sensitivity in 1999.^{3023}

Existing surface acoustic wave (SAW) devices^{458}
are chemical sensors that can measure areal mass densities as small as 80 picogram/cm^{2},
or 0.01 N_{2} molecule per nm^{2} of surface, though current
devices are limited to ~micron^{2} areas. A proposed quartz microresonator
microbalance could detect a 10^{-6 }molecular monolayer,^{459}
the equivalent of 5 x 10^{-9} molecule per nm^{2}. Detection
of ~400 million protons using an MRFM vibrating rod ~60 nm thick was demonstrated
by Rugar^{3256} in 1998.

Last updated on 17 February 2003