study of biomolecules imaging using molecular dynamics
TRANSCRIPT
Study of Biomolecules Imaging Using MolecularDynamics Simulations
Mohsen Kheirodin*, Hossein Nejat Pishkenari†,Ali Moosavi‡ and Ali Meghdari§
School of Mechanical EngineeringSharif University of Technology, Tehran, Iran
*[email protected]†[email protected]
‡[email protected]§[email protected]
Received 26 March 2015Accepted 18 May 2015Published 10 July 2015
The process of imaging a biomolecule by atomic force microscope (AFM) is modeled usingmolecular dynamics (MD) simulations. Since the large normal force exerted by the tip on thebiosample in contact and tapping modes may damage the sample structure and produce irre-versible deformation, the noncontact mode of AFM (NC-AFM) is employed as the operatingmode. The biosample is scanned using a carbon nanotube (CNT) as the AFM probe. CNTsbecause of their small diameter, high aspect ratio and high mechanical resistance attract manyattentions for imaging purposes. The tip–sample interaction is simulated by the MD method. Theprotein, which has been considered as the biomolecule, is ubiquitin and a graphene sheet is used asthe substrate. The e®ects of CNT's geometric parameters such as the CNT height, the diameter,the tilt angle, the °exibility and the number of layers on the image quality have been evaluated.
Keywords: AFM; biosample imaging; molecular dynamics simulation; CNT geometry.
1. Introduction
Due to its capability of imaging biosamples both inair and liquid at nanometer resolution, atomic forcemicroscope (AFM) has been evolved into a powerfultool in the detection of extremely small and softbiological materials. The contact mode has rarelybeen used for biosample imaging because it candamage the biomolecule structure by applying alarge repulsive force, which leads to plastic defor-mation of the biosample.1 Although the tappingmode of AFM (TM-AFM) reduces the applied force
on the sample surface compared to the contactmode, it cannot be used in imaging the soft bio-logical samples. Some repulsive interaction forcesapplied on the biosample, although fewer in com-parison with the contact mode, may a®ect the vul-nerable structure of the biomolecule. Noncontactmode of AFM (NC-AFM) does not su®er from thesample degradation e®ects, making it preferable formeasuring soft samples such as biological samplesand organic thin ¯lms. This operation mode is un-able to provide information on the mechanical
†Corresponding author.
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NANO: Brief Reports and ReviewsVol. 10, No. 7 (2015) 1550096 (12 pages)© World Scienti¯c Publishing CompanyDOI: 10.1142/S1793292015500964
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properties of the sample due to elimination of therepulsive tip–sample contact. Since in our study, weattempt to image a vulnerable protein, we will usethe NC mode.
As mentioned before, we will consider CNT asthe AFM probe. CNTs have been widely used forAFM imaging and manipulation.2–14 There arecertain properties of CNTs that make them idealtips for AFMs compared to silicon or diamond tipsincluding the ability to be fabricated with a smalldiameter ranging between approximately 1–2 nmand 0.4 nm,8 high elastic modulus, high aspect ratioand good chemical properties. Their high elasticmodulus provides a high mechanical resistancewhich helps in avoiding fracture and reducingthermal vibration e®ects even in small diameters.7
Their large aspect ratio makes them suitable forimaging a sample which has large aspect ratio too.
In spite of the signi¯cant role of tip geometry onthe direct interaction with surface and imaging res-olution, only a few papers have been devoted to thestudy of the geometry of the tip on imaging quality.For instance, Nejat and Meghdari15 investigated thee®ects of tip geometry on Amplitude ModulationAFM (AM-AFM) images using MD simulations.Their results demonstrate when the sample feature issharper than the tip, the image is dominated by theshape of the tip. In another research,16 they inves-tigated the in°uence of the system part °exibilitieson the TM-AFM measurements using MD simula-tions. They found that independent of the amplitudesetting, when the sample °exibility increases, the tipand substrate deformation and also the interactionforce decrease. Furthermore, they realized that if thesample sti®ness is small with respect to the sti®nessof the tip and the substrate (for most practicalsamples such as biological samples), then the defor-mations of the substrate and the tip are small withrespect to the sample deformation.
There are just a few works that have evaluatedthe properties of CNTs on the image quality. Forexample, Solares et al.11 investigated the e®ect oftilt angle on the imaging process of a particle withmulti-walled carbon nanotubes (MWNTs) withabout 3.5 nm diameter. They also evaluated the ef-fect of elastic deformation of single walled carbonnanotubes (SWNTs) on the imaging from a rigidnanoparticle.10
The main goal of our research is to evaluate thee®ect of geometric parameters of SWNTs with adiameter of about 0.5–1.5 nm on the sequent image
quality, then subsequently investigating the e®ec-tiveness of using double walled carbon nanotubes(DWNTs) or MWNT. Although due to their smalldiameter, it seems that SWNTs are more suitablefor imaging a small biosample with respect toMWNT, their lower sti®ness may a®ect the imagequality (as we will see).5
The method which is used here to obtain the tip–sample interaction force versus the tip height dia-gram (F–z diagram) is molecular dynamics (MD).Nowadays MD is becoming a very powerful tool formodeling dynamic behaviors of nanodevices such asAFM in imaging or manipulation processes. Forexample the modeling of the positioning process of ananoparticle with AFM,17 imaging with AFM byNejat et al.15,18–21 and Solares et al.,11,22 scratchingof a solid surface with AFM,23–25 plastic deforma-tion of a biomolecule in indentation process26 andextension a biosample with AFM27–29 have beenperformed by MD simulations to understand atomicscale behavior. In this study, we use steered mo-lecular dynamics (SMD), a feature of NAMD soft-ware, for modeling the imaging process. The nextsection explains the setup and the implementationsteps.
2. The Proposed System
In this section, we will explain the simulation setupprepared for imaging of a biomolecule in the NCmode. An AFM tip consists of two parts, the siliconcone and CNT which is located on the apex of thesilicon part. The substrate is a graphite sheet, butfor simpli¯cation in the MD simulations, we assumea graphene sheet as a substrate and calculate andadd the long-range attractive force between thesilicon part of the tip and the bulk section of thegraphite substrate according to Eq. (1), which refersto the van der Waals interaction force between acone and a sheet.30
Fflat surface�sphere ¼AHR
6D2; ð1Þ
where D stands for the nearest distance between thecone and sheet, AH ¼ 29:6� 10�20 J represents thegraphite-silicon Hamaker constant31 and R ¼ 5 nmis the sphere radius. Thus, in the MD simulations weonly consider the CNT and the graphene sheet asthe substrate and ¯nally add up the interactionforce from Eq. (1) and the MD simulation [seeFig. 1].
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We used Ubiquitin as the biomolecule, a °exiblebiosample which can be surrounded by a cube of34Å length. This protein can be found in approxi-mately all tissues of eukaryotic organisms. Thisprotein binds to other proteins and labels them fordestruction. The ubiquitin protein is composed of 76amino acids including 1231 atoms. Before using the
biosample in our simulation, it is relaxed and min-imized in a water box for 0.1 ns. Figure 2 shows theUbiquitin after minimization and equilibration.Also it should be mentioned that after merging thesystem elements together, they are minimized andrelaxed for 0.1 ns.
To reduce the biosample °exibility e®ect, we haverotated it until the most °exible part is located at thebottom of the protein. In order to prevent the lateralsliding of the protein during the approach of the tip,24 of its atoms closest to the surface were kept ¯xed.The MD setup has been illustrated in Fig. 3.
We intend to take a NC-AFM image of theUbiquitin. In this regard, the tip–sample verticalinteraction force is calculated using MD. For thispurpose, the tip approaches to the sample verticallyat a ¯xed horizontal position with a constant speed,while recording the interaction force between the tipand the surface. We want to scan a cross section ofthe protein along the x-axis.
Fig. 1. Silicon section of tip and graphite sheet interactioncalculated by long-range van der Waals relation.
(a) (b)
Fig. 2. CPK (a) and New-cartoon (b) modes representation of Ubiquitin minimization and equilibration.
Fig. 3. The prepared MD setup in horizontal and vertical views with the CNT tip, graphene as substrate and Ubiquitin asbiosample.
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In Fig. 4, the selected realistic cross section of theprotein is depicted. We divided the cross sectioninto 61 horizontal points with an equal distance of1Å. Points �30 and 61 belong to x ¼ 1:5 nm andx ¼ 7:5 nm, respectively. Then, we use MD to ob-tain for each horizontal point. In order to producean image from the surface, we calculate the inter-action force versus the image height (using Eq. (5)in Sec. 3) at these 61 points.
It is worthy to note that a realistic cross sectioncannot be captured due to several practical reasons.First of all, the geometry of the tip superimposesitself on the raw scan data measured from thespecimen. In addition, due to °exibility of the tip,the substrate and the sample, the tip–sample in-teraction force induces system deformation. Gener-ally speaking, the elastic deformation of the systemelements leads to the particle height lowering in themeasured topography. Furthermore, NC mode im-aging restricts the tip movement to the attractiveregion of the tip–sample interaction diagram. Thus,the tip position and the sequent topography areabout 3–4Å above the sample. Moreover, the vari-ety of biosample atoms and di®erent neighbor atomsof the cross section may in°uence the topography.
The best image from a sample can be taken by avery sharp (small diameter and high aspect ratio)tip having a high sti®ness (low °exibility). In orderto compare our images with this ideal image, weassume a virtual CNT tip which is rigid and has avery small diameter as the ideal tip. Then, we obtainthe biosample's cross section topography with this
tip as an ideal image. Ideal CNT is assumed to havea 3.9Å diameter, an in¯nity length and a rigidstructure. Figure 5 shows the biosample topography(red curve) with an ideal tip.
To move the tip above the biosample, we use theSMD method, which makes it possible to apply aconstant force to a number of atoms or move themwith constant velocity. This method has beenwidely used in recent years for evaluating the dy-namics behavior between the AFM tip and thesample in di®erent applications27–29,32,33 In thisstudy, we use the constant velocity SMD. In thismethod, the NAMD software moves selected atomswith speci¯ed velocity.33 As we choose higher springsti®ness, the constant velocity approximation forSMD atoms will improve.
We select 1 nm top atoms of the CNT as SMDand constrained atoms, which are forced to moveonly in the Z-direction with the SMD velocity of1m/s. Because of the multi-scale property of thesystem, simulating the tip movement with its realvelocity is very time consuming. Thus, we should¯nd an optimized velocity which does not a®ect thedynamic behavior of the system. To this end, wehave conducted several simulations by halving thetip velocity value and observing when the simula-tion results do not change considerably. Here, wehave calculated the interaction force between the tipand the sample for di®erent velocities using MD.Figure 6 depicts the (F–z) diagram for three tipswith vertical velocities of 0.5m/s, 1m/s and 2m/s.As can be seen in Fig. 6, the interaction force be-tween the tip and the protein behavior does notchange signi¯cantly near the velocity of 1m/s.
-3 -2 -1 0 1 2 30
0.5
1
1.5
2
2.5
3
3.5
X (nm)
Z (
nm)
Ideal image
cross section
Fig. 5. The image of the biosample with ideal tip, the besttheoretical image with CNTs in NC mode (color online).
1 2 3 4 5 6 7 80
0.5
1
1.5
2
2.5
3
3.5
X (nm)
Z (
nm)
C
NO
H
Fig. 4. Selected cross section of the biosample and substratewhich will be scanned by AFM probe. The selected cross sectionis divided to 61 points with an equal distance 1Å.The horizontalposition of points �30 and 61 are x ¼ 1:5 nm and 7.5 nm, re-spectively. The protein atoms depicted in ¯gure are those atomshaving a distance less than 2Å from the selected cross section.
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Thus, we choose 1m/s as a suitable speed for the tipmovement.
The biomolecule after minimization and relaxa-tion in a periodic water box, is added to the mainsetup. To prevent the biomolecule's instability andto simply model the environment, we haveemployed Langevin dynamics (LD). Although con-sidering the solvent media implicitly cannot modelthe e®ect of CNT hydrophobicity, LD method isable to mimic many e®ects of the environment byconsidering e®ects such as random and frictionalforces on the biomolecule atoms with lowercomputational cost. Other MD simulation para-meters are presented in Table 1. In our simulations,CNT atoms are supposed to be neutral, and itshould be noted that this assumption may not betrue generally and depends on the termination andfunctionalization process of the CNT.
Since the calculated F–z diagram has some un-desirable °uctuations, we use a ¯tting function toremove them. The ¯tting function used here is:
where zc2 is the tip's height corresponding to mini-mum force ðfminÞ and zc1 is the tip's height for�fmin, where the coe±cient � 2 ð0:1Þ decreases asthe CNT's °exibility increases. Other coe±cientsare obtained from the continuity of the F–z diagramand the least square method. The di®erence betweenthe proposed ¯tting function and that of Solareset al.22 are in dividing the ascending attractive re-gion into two parts, and using the exponentialfunction for the ¯rst part. Dividing the ¯rst sectioninto two subparts is necessary especially for CNTswith high °exibility which produce a (F–z) diagramthat varies a lot along the ascending attractive re-gion. Figure 7 shows the ¯tness quality of the pro-posed function at the middle point of the biosample.
Figure 8 shows the schematic lumped model ofAFM cantilever in vertical excitation (VE) imagingmodes.
Table 1. MD simulation parameters.
Variable DescriptionTemperature 300KThermostat Langevin dynamicsTime step 2 femto's, Rigid bondsBoundary Condition Graphene atoms are rigidCut o® distance 12ĂPair-list distance 13.5Ă
Fig. 8. Lumped model of AFM.
3.3 3.4 3.5 3.6 3.7 3.8-0.2
-0.15
-0.1
-0.05
0
0.05
Tip height (nm)
For
ce (
nN)
0.5 m/s1 m/s2 m/s
Fig. 6. Interaction force versus tip movement for di®erent tipvelocities, the behavior does not change in velocities below 1m/s.
For
ce (
nN)
3.2
-0.2
-0.1
0
0.1
0.2
3.4 3
Tip height (nm)
3.6 3.8 4
Fig. 7. Tip–sample interaction force curve for the system inpoint 0 (x ¼ 4:5 nm). The small circles give the MD force valuesand the solid lines correspond to Eq. (2). The proposed ¯ttingfunction are de¯ned for three di®erent regions.
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The dynamic equations of the tip motion are34:
md::ðtÞ þ bðd:ðtÞ � _xðtÞÞ þ kðdðtÞ � xðtÞÞ ¼ fðtÞ; ð3Þ
mex::ðtÞ þ bð _xðtÞ � d
:ðtÞÞ þ kðxðtÞ � dðtÞ¼ fILðtÞÞ; ð4Þ
where fðtÞ is the piezo actuator force applied to thecantilever finðtÞ and is the tip–sample interactionforce. Here, for NC mode imaging we can use two PIcontrollers to maintain the vibration amplitude andfrequency shift at the desired values by tuning fðtÞand the substrate height, respectively. Assumingperfect performance of frequency and amplitudecontrollers, the frequency shift for each vibrationcycle is given as:
�fðd; k;A; fILÞ
¼ � f02�kA
Z 2�
0
finðdþAþA cosð�ÞÞd�;ð5Þ
where A is the oscillation amplitude and k repre-sents the cantilever sti®ness. Because of small in-teraction force between the tip and the sample, thevibration resonance frequency should be large and kand A (amplitude) should be small to have a rea-sonable magnitude of frequency shifts. On the otherhand, to improve the image resolution, we shouldset the frequency shift at maximum value withinstable imaging region. We have chosen an optimizedmagnitude of frequency shift of 40Hz for our im-aging simulations. The utilized parameters in oursimulations are presented in Table 2.
3. Results
As mentioned before, to reproduce the realisticimages of the biosample, we should use two di®erentPI controllers to maintain the vibration amplitudeand frequency shift at the desired values. However,in our next simulations, we will try to measure idealimages, which can be obtained by using Eq. (5) as-suming the controllers are working perfectly. Inorder to compare the images obtained from thesetwo techniques, i.e., the perfect controller and real
controller, we have produced the biosample topog-raphy by these methods. Figure 9 depicts the e®ectof scanning velocity on the image quality. An ex-amination of the ¯gure reveals that increasing thescanning velocity reduces the image quality.
In the next sections, we attempt to investigatethe e®ects of the di®erent CNT parameters on thequality of the produced topography.
3.1. CNT 's diameter
One of the most important geometric parameters ofCNTs is their diameter. Using SWNTs makes itpossible to have tips with diameters of about 1–2 nm. While Hafner et al.,3 and Woolley et al.4 haveindicated that the minimum diameter of SWNTis about 0.7 nm, Chen et al.8 have reported theCNT diameter to be 0.4 nm.8 Generally speaking,increasing the CNT diameter leads to low°exibility, which is desirable; but because of the tipbroadening e®ect, it reduces the image quality.In the present study, we have chosen CNT dia-meters to be D1 ¼ 5:4Å, D2 ¼ 8:1Å, D3 ¼ 10:9Åand D4 ¼ 14:1Å with a length of L1 ¼ 50Å. Thetip broadening e®ect for di®erent CNTs is depictedin Fig. 11. The measured topography illustrated inFig. 11 are obtained assuming that all of the systemelements are rigid. This assumption help us to dis-tinct the e®ects of the tip broadening from thee®ects of the tip °exibility (Fig. 12). From theresults shown in Fig. 10, it may be concluded thatthe tip broadening e®ect can be easily calibrated.
Figure 11 shows the biosample topography forthe real °exible tip and biosample. Results indicatethat as the tip diameters increases the image's
Table 2. The vibration's parameters.
A, Amplitude 11 nmk, Sti®ness 2N/mf0, Resonance frequency 200 kHzDesired frequency shift 40 z
-4 -3 -2 -1 0 1 2 3
1
1.5
2
2.5
3
3.5
4
X (nm)
Z (
nm)
V=100 m/sV=400 m/sV=1600 m/sEq. (5)
-2.1 -2.05 -2 -1.95 -1.9 -1.85 -1.8 -1.75 -1.7
1
1.5
2
2.5
3
Z (
nm)
X (nm)2 2.1 2.2 2.3 2.4
1
1.5
2
2.5
Z (
nm)
X (nm)
Fig. 9. The e®ect of the scanning velocity on the imagequality, as the scanning velociy increases, the overshoot of themeasured topography with respect to ideal image increases.
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quality improves due to the smaller tip °exibility,but this e®ect is less than the tip broadening e®ect.The e®ect of tip's length and °exibility will be in-vestigated more in the next section. As we men-tioned previously, the ideal image shown in Fig. 5 isthe image taken by a rigid CNT with a diameterD1 ¼ 3:9Å having an in¯nity length. As is evidentin Fig. 11 the measured images are wider than theideal image because of the tip broadening e®ect andtheir °exibility. Furthermore, the in¯nity length ofthe ideal image leads to zero force between the sili-con part of the tip and the graphite. Thus, the idealimage is below the other images.
3.2. CNT 's length
It is expected that if the CNT length increases theimage quality decreases, due to more tip de°ection.The e®ect of CNT length on its °exibility and
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
tip movement (nm)
tip
defl
ecti
on (
nm)
L1L2L3L4L5
(a)
L1L2L3L4L5
0 1 2 3 4 5-0.1
0
0.1
0.2
0.3
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0.5
0.6
tip movement (nm)
tip
defl
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on (
nm)
(b)
L1L2L3L4L5
0 1 2 3 4-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
tip movement (nm)
tip
defl
ecti
on (
nm)
(c)
Fig. 12. The e®ect of CNT's length on its de°ection in x (part(a)), y (part (b)) and z (part (c)) directions. As the CNT lengthincreases its deformation increases too.
-3 -2 -1 0 1 2 30
0.5
1
1.5
2
2.5
3
3.5
4
X (nm)
Z (
nm)
D1D2D3D4cross section
Fig. 10. The tip broadening e®ect. The image becomes broaderas the CNT diameter increases. In this simulation, the systemelements are assumed to be rigid.
-3 -2 -1 0 1 2 30
1
1.5
2
2.5
3
3.5
X (nm)
Z (
nm)
D1D2D3D4cross sectionIdeal image
0.5
Fig. 11. The e®ect of the tip diameter on the image quality.In this simulation, all the system elements are assumed to be°exible.
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de°ection has been illustrated in Fig. 12, showingthe tip de°ection in di®erent directions (for pointþ 1). The CNT lengths are assumed to beL1 ¼ 30Å, L2 ¼ 40Å, L3 ¼ 50Å, L4 ¼ 60Å andL5 ¼ 70Å with a diameter of D1 ¼ 5:4Å. Inten-tionally the tip's diameter is considered to be smallto demonstrate the CNT °exibility e®ects moreevidently.
The e®ect of CNT length on the interaction forcefor a middle point and an edge point of the bio-sample is depicted in Figs. 13 and 14.
The CNTs shown in Figs. 13 and 14 indicate theconsiderable tip de°ection due to interaction forcesbetween tip and sample. Therefore, it is reasonableto conclude that images taken by longer CNTs havemore topographical irregularities than the images ofshort CNTs. Figure 15 shows the e®ect of CNT'slength on the biosample topography and it is obvi-ous that as the CNT's length increases the image'squality decreases, in particular for the edge points.
As mentioned before, as the CNT lengthdecreases the distance between the silicon part ofthe AFM tip and the substrate decreases. In fact, inthese conditions the van der Waals part of the in-teraction force, between the silicon part of the AFMtip and the substrate, has a substantial role on thetotal force exerted on CNT. This force, between thebulk parts of the tip and the substrate, reduces thee®ects of the local interaction forces, betweenatomic parts of the tip and the sample, on themeasured topography leading to a lower image res-olution. When the CNT length increases, the bulkforces between the tip and the substrate reduces andto produce the desired frequency shift the tip shouldapproach closer to the substrate. This e®ect is ob-vious in Figs. 15 and 16. Since the ideal tip has an
-3 -2 -1 0 1 2 30
0.5
1
1.5
2
2.5
3
3.5
X (nm)
Z (
nm)
L=3 nmL=7 nmL=20 nmcross sectionIdeal image
Fig. 16. The e®ect of CNT's length on image's quality forD ¼ D5, in this case °uctuations appear in length of 20 nmbecause of higher CNT diameter.
3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3
-0.15
-0.1
-0.05
0
Tip height (nm)
For
ce (
nN)
L1L2L3L4L5
Fig. 13. The e®ect of CNT's length on F–z diagram, point þ 1(middle).
1 1.5 2 2.5 3-0.2
0
0.2
0.4
0.6
Tip height (nm)
For
ce (
nN)
L1L2L3L4L5
Fig. 14. The e®ect of CNT's length on F–z diagram, pointþ 22 (edge).
-3 -2 -1 0 1 2 30
0.5
1
1.5
2
2.5
3
3.5
X (nm)
Z (
nm)
L1L2L3L4L5cross sectionIdeal image
Fig. 15. The e®ect of CNT's length on image's quality forD ¼ D1, higher length cause to more °uctuations in image andlower long range attractive force.
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in¯nity length, the image taken by this tip is belowthe other images.
Now we examine the CNT length e®ect for alarger diameter D5 ¼ 12:51Å. As expected, theresults indicate that the CNT °exibility exposes itse®ect on the images quality at larger lengths. Herewe examine the CNT lengths of L1 ¼ 30Å,L5 ¼ 70Å and L6 ¼ 200Å. As is illustrated inFig. 16, the image corruption, because of CNThigher °exibility at higher lengths, occurs atL6 ¼ 200Å and not at L5 ¼ 70Å.
3.3. CNT 's tilt angle
One of the imaging artifacts, which may createproblems in the imaging, arises from the probe tiltangle. Here, we examine the e®ects of CNT tiltangles in the x-direction (perpendicular to scanningdirection) and y-directions on sequent images. Thetilt angles were supposed to be þ 20�, þ 40�, �20�and �40� in the x-direction and þ 20� and þ 40� inthe y-direction. The ¯rst e®ect is that because of tipapex deviation, the tip–sample interaction is in°u-enced by the neighboring atoms of the sample ratherthan the desired atoms (atoms placed exactly belowthe CNT), this e®ect can be named \shape e®ect ofthe CNT tilt" and can be seen obviously in Fig. 17,whose images are achieved with rigid atoms as-sumption. Unfortunately, despite the tip broaden-ing e®ect, this shape e®ect error cannot becalibrated. The di®erence between the measuredand the ideal image increases with increasing the tilt
angle. The second problem is that the CNT tiltcauses a bending force along the CNT, which leadsto the tip de°ection. Solares et al.11 investigated themechanical e®ects of tilt angle in SWNT andMWNT tips during tapping mode imaging from arigid nanoparticle.
Figure 18, which shows the e®ect of tilt angle onthe image quality with MD simulations, expressesthat this e®ect is negligible compared to the shapee®ect. The CNTs have a diameter ofD1 ¼ 5:4Å anda length of L3 ¼ 50Å.
3.4. CNT 's °exibility
The purpose of this section is the comparison be-tween the e®ect of the CNT and biosample °exi-bility on the measured topography of the biosample.Generally speaking, as the CNT aspect ratiodecreases or number of its layers increases, the in-°uence of CNT °exibility increases. Shapiro et al.10
studied the e®ect of SWNT and sample deformationduring the imaging process. Here, we compare the°exibility e®ect between a CNT with diameterD1 ¼ 5:4Å and length of L3 ¼ 50Å and ubiquitinas the biosample. First, we obtain the image as-suming that all of the system elements are °exible.Second, we consider the biosample atoms rigid.Third, consider just the CNT atoms rigid and ¯nallyconsider the whole setup atoms rigid and comparethe obtained topographies from the four mentionedconditions. Figures 19 and 20 show the comparisonof (F–z) diagrams for the four conditions for amiddle point and an edge point of the biosample.
As we expected the trend of F–z diagrams forboth edge and middle points are similar to the (F–z)
-3 -2 -1 0 1 2 30
0.5
1
1.5
2
2.5
3
3.5
X (nm)
Z (
nm)
-40 deg-20 deg0 deg20 deg40 degcross sectionIdeal image
Fig. 18. The e®ect of CNT's x-direction tilt on image's quality.
-3 -2 -1 0 1 2 30
0.5
1
1.5
2
2.5
3
3.5
X (nm)
Z (
nm)
-40 deg-20 deg0 deg20 deg40 degcross sectionIdeal image
Fig. 17. The the e®ect of tilt angle on biosample's image,atoms are considered rigid, this error despite of tip broadeningcannot be calibrated.
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diagrams for di®erent lengths (Figs. 13 and 14).Figure 21 shows the obtained topographies. It canbe concluded that the e®ect of biosample °exibilityis more than the e®ect of CNT °exibility and this isespecially obvious for edge points.
3.5. MWNTs or SWNTs
It is expected that SWNTs, because of their lowerpossible diameters, are more suitable for moleculescale imaging than MWNTs. On the other hand,DWNTs may improve imaging quality in some casesbecause of their mechanical sti®ness is more com-pared to SWNTs.4 Here we consider two SWNT tipswith diameters of 5.4Å and 15.2Å and length 70Å.Also we use a DWNT with inner diameter of 5.4Å,outer diameter of 15.2Å, and length 70Å. In
manufacturing MWNTs, the gap distance betweenthe tubes must be at least 3.4Å.4
It is worthy to note that, two SWNT tips are thesame as inner and outer tubes of DWNT tip, re-spectively. We name the small SWNT as inner tube,and the large SWNT as outer tube. Figure 22depicts the image taken by two SWNT tips andDWNT tip. It is obvious that the image taken bythe inner tube is more accurate than the ones takenby the outer tube and DWNT. However, theDWNT and outer tube calibrated images will bemore accurate because of their lower aspect ratioand °exibility.
As it is shown in Fig. 22, the measured topogra-phy of the outer tube (SWNT) and DWNT are
-3 -2 -1 0 1 2 30
0.5
1
1.5
2
2.5
3
3.5
X (nm)
Z (
nm)
all rigid
protein rigid
CNT rigid
none rigid
cross section
Ideal image
Fig. 21. The e®ect of CNT's and biosample's °exibility onimage's quality, the sample °exibly has the greatest role inimage corruption.
3.4 3.6 3.8 4 4.2 4.4-0.2
-0.15
-0.1
-0.05
0
Tip height (nm)
For
ce (
nN)
non rigidprotein rigidtip rigidwhole rigid
Fig. 19. The e®ect of CNT's and biosample's on F–z diagram,point þ 1 (middle), the trends are like the F–z diagram fordi®erent lengths.
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0.5
1
1.5
2
2.5
3
3.5
X (nm)
Z (
nm)
DWNT
inner tube
outer tube
cross section
Ideal image
Fig. 22. Comparison between the obtained image with SWNTand DWNT, outer tube and DWNT have about the sameimages for L ¼ 7 nm.
0.5 1 1.5 2 2.5 3-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
Tip height (nm)
For
ce (
nN)
non rigidprotein rigidtip rigidall rigid
Fig. 20. The e®ect of CNT's and biosample's on F–z diagram,point þ 22 (edge), the trends are like the F–z diagram for dif-ferent lengths.
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approximately similar. To study the role of the tiplength on the image taken by these tips, we haveincreased the tips length from 70Å to 200Å. Fig-ure 23 illustrates the images taken by two SWNTtips as well as the image taken by DWNT tip. As itis demonstrated in this ¯gure, the image of DWNTis more precise than the images taken by SWNTs.The images of SWNTs have more °uctuations par-ticularly in the edge points of the sample in com-parison with the image of DWNT and this is due toadditional °exibility of the SWNT tips.
To examine the sti®ness of the mentioned tips,here we have compared the °exibility of the largerSWNT with the °exibility of DWNT. For thispurpose, we have ¯xed one side of the CNTs andapplied a constant force F ¼ 0:2 nN perpendicularlyon the tip of CNT. The results indicate that thesti®ness of the DWNT tip is about 10% more thanthe sti®ness of the SWNT. This is the reason whythe DWNT image has less °uctuations compared tothe SWNTs images.
4. Conclusion
Characterization of the nanoscale biosamples withhigh spatial resolution, is an essential technology forthe study of biological materials. In this paper, wepresented the simulation of biosample imaging byNC-AFM. The results provide an important physi-cal insight, which reveals the signi¯cant role of thedi®erent parameters such as the diameter, length,number of layers (DWor SW) and tilt angle of the tip
on NC-AFM measurements. Some of the main con-clusions demonstrating the e®ects of the tip para-meters on NC-AFM measurements are as follows:
It has been demonstrated that some of para-meters such as the tilt angle, diameter and length ofthe tip a®ect the imaging quality signi¯cantly whilethe other parameters such as the number of layers ofthe tip and the scan speed have a little e®ect on themeasured topography.
Independent of the frequency shift set-point, theimages taken in the NC-AFM mode with the as-sumption of a rigid tip and sample is above theimage of the °exible realistic system, and the dif-ference between these two images increases in more°exible parts of the sample.
The size of the CNT diameter has two e®ects onthe image quality. First, the tip broadening causedby the tip's shape and, the second, the e®ect of°exibility. The e®ect of the tip broadening can becalibrated easily, and the °exibility at a length lessthan 50Å has a little e®ect on the image quality. Weexamined CNTs with di®erent lengths of 30Å to70Å. Results revealed that as the CNT lengthincreases the image quality decreases, particularlyin the edge points. Of course this e®ect is negligiblefor larger diameters in this range of length. For in-stance, the results demonstrate that for a CNT withthe diameter of D ¼ 12:5Å, the CNT length onlyin°uences the image quality of L200Å.
Similar to the tip diameter, the tip tilt angle hastwo e®ects on image quality: a mechanical e®ect anda shape e®ect. In our scales, the mechanical e®ectsare negligible compared to the shape e®ects. Fur-thermore, despite the tip broadening e®ect which isthe shape e®ect due to the nonzero CNT diameter,the shape e®ect of CNT tilt angle on achievedimages cannot be calibrated.
Although, DWNTs are sti®er than SWNTs, thisissue has a small e®ect on the image quality whenCNT length is lower than 10 nm (aspect ratio ofabout 4–5). But, DWNTs exhibit more sti®ness andlower de°ections when CNT length exceeds 20 nm,where the image of SWNTs has too °uctuations anderrors, especially in the edge points.
A series of simulations have been conducted tocompare the e®ect of the biosample °exibility on themeasured topography with the e®ect of the CNT°exibility on it. The results showed that althoughthe examined CNT (D ¼ 5:4Å and aspect ratio of10) is too °exible, the °exibility of biosample plays amore important role on the resultant image.
-3 -2 -1 0 1 2 30
0.5
1
1.5
2
2.5
3
3.5
X (nm)
Z (
nm)
SWNTDWNTIdeal imagecross section
Fig. 23. The image of a monolayer and a double layer CNTwith L ¼ L6, for length of 20 nm the higher mechanical resis-tance of DW shows itself.
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Acknowledgment
The authors would like to thank the Iranian Na-tional Science Foundation (INSF) for their ¯nancialsupport.
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