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

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

4.4.1 Minimum Detectable Force

Extremely small forces have been measured using macroscale instrumentation, particularly the atomic force microscope (AFM). In 1998, experimental forces of 1 nN in the contact regime and 1 pN in the noncontact regime were routinely measured.⁴⁵⁰ In 1994, D. Rugar achieved a force sensitivity of 5 x 10^-16 N using an AFM with a 90-nm thick cantilever at room temperature,⁴⁸⁹ and in 1998 Rugar reported a force resolution of 7 x 10^-18 N using a magnetic resonance force microscope (MRFM) with a 230-micron long, 60-nm thick, silicon cantilever.³²⁵⁶ The theoretical limit on the force sensitivity of the AFM has been estimated as 10^-18 - 10^-19 N for temperatures near absolute zero^445,489 using fiber optic interferometry or other techniques. However, many of these methods are impractical or impossible to employ in medical nanodevices.

What is the minimum force F_min likely to be detectable by nanorobot sensors? Several lines of reasoning produce consistent answers. For example, force may be measured by the displacement Dx of a beam against the known resistive pressure provided by a spring of minimum stiffness k_s; that is, F_min ~ k_s Dx. However, the energy stored in the spring must exceed thermal noise energy, with SNR ~ ln (k_s Dx² /2 kT) (Eqn. 4.9), giving F_min = 2 kT e^SNR / Dx. For T = 310 K and SNR = 2, F_min = 100 pN at Dx = 0.6 nm, 10 pN at 6 nm, and 1 pN at 60 nm. Indeed, the natural molecular motors (kinesin, dynein, and actin-myosin) have isometric force outputs on the order of 1-14 pN per protein molecule, which are routinely observed using microscale cantilever spring sensors measuring total forces of 10-100 pN.^{452-455,1058,1247} Breaking antigen-antibody or receptor-ligand interactions generally requires rupture forces in the 50-300 pN range.³²⁰⁰ A force of ~160 pN separates biotin that is seated in its avidin receptor; ~85 pN pulls iminobiotin out of the avidin receptor.¹⁰⁷⁵ A force of ~300 pN ruptures a hydrogen bond between two isolated water molecules,¹⁰³⁰ but >3000 pN are likely to be needed to rupture a single C-C covalent bond.^10,1131 The rupture strength of single covalent bonds has been directly measured experimentally by AFM as, for instance, 2000 ± 300 pN for Si-C (528 zJ/molecule dissociation enthalpy, 0.185 nm bond length) and 1400 ± 300 pN for Au-S.³¹⁹³

Force sensing may also require using measurement devices that involve sliding interfaces of two mutually inert periodic surfaces. This motion may be characterized by the potential energy of interaction, ~0.1-1 zJ as determined by Monte Carlo simulations of typical diamondoid material at 0.2-0.5 nm separations.¹⁰ Assuming a minimum 0.5 zJ energy barrier, for small sliding contacts the typical force amplitude F_min ~ 1.7 x 10¹⁰ DV_barrier ~ 8.5 pN.¹⁰

A force of 10 piconewtons across a small-molecule receptor (0.3 nm)³ of surface area 0.09 nm² represents an energy density of ~10⁸ J/m³ x 0.027 nm³ = 2.7 zJ ~ 0.6 kT. And the nanorobotic manipulator arm described in Section 9.3.1.4 is capable of ~picometer step sizes, representing a minimum applicable force of F_min ~ 10 pN assuming 0.04 m/N compliance¹⁰ for the entire mechanism. Thus as a conservative estimate, the smallest force that is reliably detectable (and applicable) by nanodevices is probably ~10 pN.

Finally, a nanosensor of dimension L able to distinguish two forces differing by DF newtons has a probability of erroneous measurement¹⁰ of

{Eqn. 4.23}

For DF = 10 pN and T = 310 K, P_err = 1% for L = 8 nm, or 0.01% for L = 16 nm, so reliable nanoscale picoforce sensors may be ~10-20 nm in size.

Last updated on 17 February 2003