Proceedings of the 2009 4th IEEE InternationalConference on Nano/Micro Engineered and Molecular Systems

January 5-8, 2009, Shenzhen, China

Design and Analysis of Helical Flagella Propelled Nanorobots

Subramanian S.I, J. S. Rathore1, andN. N. Sharma1

,*

IMechanical Engineering Department, Birla Institute ofTechnology and Science, Pilani, India

Abstract - Advancement in the field of nanorobotics has beenfacilitated by the current advances in nano-bio-technology andnanofabrication methods. The important uses of nanorobots arein advancing medical technology, healthcare and environmentmonitoring. In bio-medical applications, nanorobots need toswim in biological fluids flowing in narrow channels of fewhundred nanometer size. The dominating effects in nanometersize domains are increased apparent viscosity and low Reynoldsnumbers which makes the design of a propulsion mechanism achallenging task. Micro and nano size biological organisms moveby generating planar waves or rotating helical flagella. In thepresent work, design of propulsion with helical flagella isproposed and a generalized analytical model is developed,simulated and discussed. The performance parameters of thedeveloped model viz. velocity and efficiency have been computedbased on resistive force theory and compared with those of themodel available in literature. Improved performance, feasibilityand generality of the developed flagellar model have beendiscussed.

Keywords - Flagellar hydrodynamics, Helical flagellarpropulsion, Nanorobotics, Resistive Force Theory

I. INTRODUCTION

Miniaturization has many appealing advantages and hasattracted a lot of attention with improvements in enablingtechnologies like VLSI, MEMS and Nanotechnology. Smallsize of the order of few nanometers to a few micrometers isitself advantageous, because of being comparable to somebiological dimensions. Miniaturized robots can be used forvarious tasks where the main challenge has been the size. Themost important of the possible uses ofnanorobots is in the fieldof medicine where it has the potential to revolutionize themethods of diagnosis and treatments for diseases like cancer.The existing methods like radiation therapy are not specific andhave effects on other non cancerous cells of the body. Forsimilar applications, nanorobots have to move through bodyfluids flowing in narrow channels of the body. The motion ofnanorobots in narrow channels is eclipsed by an increase inapparent viscosity, low Reynolds number and negligible inertiacompared to viscous effects. The important qualities desirablefor nanorobots are high velocity, efficiency, specificity,controllability and a simple propagation mechanism that can berealized with miniaturized parts. In the realization ofnanorobots, other challenges are wearing and friction,significant Brownian motion due to thermal agitation and nonrigidity in nano domains due to which it is more propitious toswim or fly compared to crawling and walking [1]. Inspired bythe fact that microorganisms existing in nature functionexpeditiously under these circumstances, scientists andengineers have shown a great interest to conceptualize, model,analyze and make micro-sized swimmers that can move inbody fluids [1-5]. Efficiency ofpropulsion is extremely low but

*Contact author e-mail: nns@bits-piianLac.in

is not a cause of concern considering the abundance of energyavailable to the micro and nano sized biological organisms [2].Considering an artificial nanorobot, energy available on boardand efficiency of propulsion are critical for practical uses ofthese nano devices. Two major approaches to model propulsionin nano domains are Resistive Force Theory (RFT) and SlenderBody Theory (SBT). The SBT is more accurate, but it iscomputationally intensive compared to RFT [3]. RFT has beenused in published literature to model and analyze promisingpropulsion mechanisms achieved by planar or helical cilial andflagellar motion. Gray and Hancock [4] developed a generaltheory for the propulsion of spermatozoa based on resistiveforces generated due to planar bending waves. In [6], Honda etale proposed a new method for propulsion by rotating aferromagnetic helical coil using magnetic field. Despite thelimitations such as low velocity, it circumvents the problem ofincorporating an efficient motoring mechanism within thehead. Biomimetic propulsion mechanism was considered byBehkam et ale [7], where the performance has been estimatedby modeling the dynamics of motion. Inspired by the motilitymechanism ofprokaryotic and eukaryotic microorganisms withperitrichous flagellation, models for the propulsion ofmicrorobots have been developed and validated using a scaledup model [8].

In the planar bending wave model, elements of the flagellaundergo opposite strains often and energy has to be generatedlocally for sustaining the same amplitude throughout the lengthof the tail. The reversible straining is less in case of helicalflagellar propulsion mechanism. Behkam et ale [7-8] presenteda Biomimetic propulsion model by using RFT for analyzing theperformance of a uniform amplitude helical flagellum. Theflagellum is considered as a long thin (··..,20 nm thick), helix (2.5Jllll pitch, 0.5 Jllll diameter) that rotates at a speed of,....,100 Hz.The helix of the flagellum is considered to have a constantamplitude and constant pitch in [7-8]. The influence of varyingamplitude and pitch needs to be explored for improvedperformance of the propulsion mechanism. In the present work,a generalized case of helical wave mechanism is modeledwhere the amplitude (y) varies along the length of the flagella.The rotation of the helical flagella about the longitudinal axis,results in a net forward thrust, in viscous domains, whichpropels the robot forward. The mathematical model for theforces, velocity, and efficiency, considering a general case ofthe helical flagellar mechanism, is developed. The investigationon performance parameters for the linearly varying amplitudehelical flagellar mechanism is done and is compared with theconstant amplitude model. The results are discussed forimproved efficiency of propulsion for the nanorobot. Theresults obtained show that there is a scope for significantimprovement in performance by varying the amplitude andpitch. The present work is organized into three more sections.

978-1-4244-4630-8/09/$25.00 ©2009 IEEE 950

Authorized licensed use limited to: TATA INSTITUTE OF FUNDAMENTAL RESEARCH. Downloaded on June 6, 2009 at 03:13 from IEEE Xplore. Restrictions apply.

Fig. 1 shows the schematic of the nanorobot with a variableamplitude helical flagellum and an inert head. The diameter ofthe head is denoted as 2a, the thickness of the filament is 2d,the uniform pitch of the helix is p and the amplitude, y variesalong the longitudinal axis, x. A small element of the helix isshown in Fig. 2 where, y is the radial axis, x is the longitudinalaxis and the length of the element is ds. The instantaneouspitch angle, (), is given by

Fig. 1. Schematic ofa nanorobot with varying amplitude helical flagellum

tane = 2JZYP

From Fig. 2 the length ofthe element is obtained as,

ds = l+cos2 e(dy)2~dx cose

(1)

(2)

In the next section, methodology and modeling of theproposed mechanism is presented. In section III, simulationand significant results of the developed model are discussed. Inthe last section, conclusions and scope for future work arementioned.

The angle between ds and dl is very small when thegradient of the amplitude is small and the tangential componentof the drag force, JL, acting on the element is approximated as

(3)

and the normal component ofthe drag force, IN, is equal to

(5)

(6)

II. METHODOLOGY AND MODELING

Surface area to volume ratio increases with reduction insize and surface effects become prominent in nano domains.Dimensions of the nanorobot depend on its applicationrequirements and the capability of the available fabricationtechnology. Small dimensions of the robot and the nature offlow in small channels results in very low Reynolds numbers.Viscous effects dominate the inertial forces and the flow istermed as Stokes flow. Models for nanorobot propulsion havebeen developed considering the conditions of dominatingviscous forces, low Reynolds number and negligible inertia.

The hydrodynamics of flagella are analyzed using RFT tomodel the propulsion of nanorobots. The components of dragforce that are generated depend on the components of thevelocity along the respective directions. The developed modelof nanorobot propulsion is valid for low Reynolds numberswhere inertial forces are negligible.

y

(4)

where Vx is the forward velocity and Ve is the equivalenttranslational velocity of the element resulting from the rotationabout the longitudinal axis. CL and CN are the coefficients ofresistance or the drag coefficients along the tangential andnormal directions which relate the velocity and the drag forcealong the respective directions. The expressions for CL and CN

for a single flagellum with free ends are given by [3]

C - 21lJ1

L - m(2:)_~C - 41lJ1

N - me:)+~From equations (1) through (4), the forward thrust due to

the small element is obtained as

Under steady state conditions, the magnitude of the thrustforce generated by the entire helical tail is equal to the dragexperienced by the head:

The motor mechanism inside the head rotates the shaft withan angular velocity w. For the resultant torque applied on therobot to be zero, the head rotates in the opposite direction withan angular velocity n [8]. Therefore, the helical filamentrotates with an absolute angular velocity of Wact = W - n .

(8)x

dx,,,,,

, 'dl', ", ,

'Ilf

zFig. 2. A small element ofthe varying amplitude helix

951

Authorized licensed use limited to: TATA INSTITUTE OF FUNDAMENTAL RESEARCH. Downloaded on June 6, 2009 at 03:13 from IEEE Xplore. Restrictions apply.

Considering only small gradients in amplitude, the torquerequired to rotate the small element shown in Fig. 2 is given as

The drag experienced by a spherical head of radius a, in amedium with viscosity f.l is given as

(Log Senlt"

-510

Radus of head (m)

0.014

0.012-3

0.01 x 10

~ 0.008 112

c::.~ 0.006 10

~0.004 ·8

0.002- 6

0

0

I~

(9)

(10)

(11)

Fhead = 6111LaVx

and the torque required to rotate the head is given by

M head =8111La3n

The steady velocity of the nanorobot, Vx, is calculated fromequations (5) through (8) and (10). Efficiency of the system iscomputed as the ratio of the useful power developed (for axialmotion) to the total input power required for spinning the headand the flagella:

(12) Fig. 4. Variation of efficiency with head size and (Q/wact) for the varyingamplitude helical model in a medium with J.l=4mPa.s

III. SIMULATION AND RESULTS

The simulations of the developed efficiency and forwardvelocity models are done in MATLAB considering dimensionsand angular velocity of the helix as user inputs. A flagellumwith 20 turns is considered for simulations. The viscosity ofblood through which the nanorobot is assumed to propagate isconsidered as 4 mPa.s. A Number of helical profiles areconsidered for flagellar design; the two profiles with constantand linearly varying amplitude are presented and discussed inthe present work.

Fig. 5. Comparison between the efficiency of constant and varying amplitudehelical models with the optimum head size, in a medium with J.l = 4mPa.s

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

o/Waet

For the constant amplitude helical model, the variation ofthe efficiency with respect to size of the head and the ratio ofthe angular velocity of the head to that of the tail is obtainedand plotted in Fig. 3. It is observed from the plot in Fig. 3, thatthe efficiency has a peak corresponding to an optimal head sizefor the condition of a stationary head (Q/ffiact = 0). Efficiencyvaries with changing n/ffiact. The optimal head size formaximum efficiency is a function of (n/ffiact). The varyingamplitude model is presented for which the amplitude is

(13)y=2xM/np

6

I::- - 8

(LogScnlt"Radius of head (m)

-410-3

10

oo

0.02

0.01

~i'uIEw 0.005

0.015

Fig. 3. Variation of efficiency with head size and (Q/wact) for the constantamplitude helical model in a medium with J.l = 4mPa.s

where x is the distance from the head along the longitudinalaxis, np is the total length along the longitudinal axis and M isthe mean amplitude of the profile. The amplitude varies fromzero and approaches its maximum value and the number of

952

Authorized licensed use limited to: TATA INSTITUTE OF FUNDAMENTAL RESEARCH. Downloaded on June 6, 2009 at 03:13 from IEEE Xplore. Restrictions apply.

oL--....L..-.L...-..l..-L....L.J....L.....L---'----J...-1---'--"-I...L...L..l.------'--"'---'--'--'-..............L_"'-------'----l..-.L...L...L.J....L.L-~::l!:!!:l~...

10-8 10.7 10.6 10-5 10.4 10.3

Radius of head (m) (log scale)

Fig. 6. Comparison of the propagation velocity in constant and varyingamplitude helical models in a medium with J.l = 4mPa.s

REFERENCES

[1] N. N. Sharma and R. K. Mittal, "Nanorobot movement: Challenges andbiologically inspired solutions," in Proc. 4th Int. Conf onComputational Intelligence, Robotics and Autonomous Systems,Palmerston North, New Zealand, 2007, pp. 19-25.

[2] E.M. Purcell, "Life at Low Reynolds Number," American Journal ofPhysics, vol. 45, pp. 3-11, 1977.

[3] Johnson R. E. and Brokaw C. J., "Flagellar Hydrodynamics: Acomparison between resistive force theory and slender body theory,"Biophysical Journal, vol. 25, pp. 113-127, 1979.

[4] Gray, J., and G. J. Hancock, "The propulsion of sea-urchinspermatozoa," Journal of Experimental Biology, vol. 32, pp. 802-814,1955.

[5] Brokaw C. J., "Flagellar Propulsion," J. Exp. BioI., vol. 209(6), 2006,pp. 985-986.

[6] Honda, T., Arai, K., and Ishiyama, K., 1999, "Effect of Micro MachineShape on Swimming Properties of the Spiral-Type Magnetic Micro­Machine," IEEE transaction on magnetics, vol. 35, pp. 3688-3690.

[7] B. Behkam and M. Sitti, "Modeling and testing of a biomimeticflagellar propulsion method for microscale biomedical swimmingrobots," in Proc. Int. Conf on Advanced Intelligent Mechatronics, IEEE/ASME, Monterey, California, 2005, pp. 37-42.

[8] B. Behkam and M. Sitti, "Design methodology for biomimeticpropulsion of miniature swimming robots," Journal of DynamicsSystems, Measurements, and Control, Transactions of the ASME, vol.128,pp.36-43,2006.

[9] R. Majumdar, J. S. Rathore and N. N. Sharma, "Simulation ofswimming nanorobots in biological fluids," in Proc 4th Int. Conf onAutonomous Robots and Agents, New Zealand, 2009 (accepted).

viscosity and can potentially be used as a trigger in applicationssuch as sensing and drug delivery [9].

The work may be extended to search and optimization ofdesign parameters for increased efficiency in nanorobotpropulsion. The model can be further improved by consideringinter-flagellar interactions and its implications.

'\\\~\-.

\".\

"~'"

--\.'.:~,

5 X 10-5

0.5

_ 3.5

i~ 3Z.

~ 2.5>6:;; 2o

~a.. 1.5

turns, length of filament and the mean amplitude are kept thesame as in the case of the constant amplitude helical flagellum.

The efficiency variation for linearly varying helical profileflagellar propulsion mechanism is shown in Fig. 4. Theassumption of gradual increase in amplitude is valid for thelinearly varying amplitude, with the angle between ds and dlbeing less than 0.58°. The nature of the plot is similar to thatobtained for the constant amplitude case. Fig. 5 shows thevariation of the efficiency with the angular velocity ratio forthe optimum value of head size (--3.98 Jlll1) for the twoconsidered cases of constant and linearly varying amplitudehelix. The efficiency of the varying amplitude case is higher forn!roact > 0.047 and the trend is opposite for the values lesserthan 0.047. The propagation velocity is shown as a function ofthe head size for both the constant and variable amplitude helixin Fig. 6. The propagation velocity for the varying helix profileconsidered in the present work is higher than that for theconstant amplitude helix. An increase in the viscosity leads toan increase in the thrust force but is balanced by an increase inthe drag of the head. Therefore, variation in viscosity does notresult in any change of parameters like velocity and efficiency.The model is valid only for the assumptions ofviscous flow.

IV. CONCLUSION

An analytical model for propulsion attributed to generalizedhelical flagella has been developed. Keeping the meanamplitude constant, the linearly increasing amplitude helicalmodel showed improved performance compared to the uniformamplitude model. Thus, a plausible design parameter forincreased propagation of nanorobots is the helix profile. Apartfrom increment in velocity of nanorobot, the linear variation inamplitude reduces internal strains which may develop at thebasal end of the flagella for constant amplitude helix.

Performance characteristics are a function of onlydimensions and geometry of the helical flagella, where as, thethrust force experienced by the nanoswimmer changes with

953

Authorized licensed use limited to: TATA INSTITUTE OF FUNDAMENTAL RESEARCH. Downloaded on June 6, 2009 at 03:13 from IEEE Xplore. Restrictions apply.