Принципы нанометрологии (1027506), страница 43
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To convert the cantileverdeflection measured by the photodiode in volts to metres, a displacementconversion factor (also called the optical lever sensitivity) is obtained fromthe region of the force curve where the sample is in contact with the cantilever. For an infinitely hard contact, every displacement of the piezoelectricscanner displaces the sample or the tip; the cantilever is pushed upwards,which is recorded as a voltage output on the photodiode.
The slope of theforce curve in the region where the cantilever is in contact with the sampledefines the optical lever sensitivity. This part of the force curve is called theregion of constant compliance or region of contact.It is important to note that using the constant compliance region of theforce curve to convert photodiode response to deflection will overestimatethe force of interaction if the cantilever is not the most compliantcomponent of the system. This is often the case when soft, deformablesubstances such as polymers are used in force measurements (either asa sample or linked to the tip/cantilever). If a compliant substrate is used,other methods are needed to accurately convert the measured deflection ofthe cantilever into a force of interaction [31].
In this case the optical leversensitivity is determined by pressing the tip/cantilever against a hardsample (for example, mica), before and after it is used on a soft sample.However, often this method does not work as the optical lever sensitivity isstrongly dependent upon a number of factors.
These factors include theposition and shape of the laser spot and the difficulty in precisely aligningthe laser spot on the same position on the cantilever from experiment toexperiment. Also, the use of a hard sample cannot be applied if it is the tip/cantilever that supports the most compliant component of the system (forexample, a molecule attached to the cantilever). Another method thatrelies on the ‘photodiode shift voltage’, a parameter that is very sensitive tothe position and shape of the laser of the photodetector, can be used toconvert volts of cantilever deflection into metres of deflection [32]. Thismethod ensures that forces can be determined regardless of the complianceof the cantilever relative to any other component in the AFM, and alsoensures the preservation of fragile macromolecules, which may be presenton the sample or attached to the cantilever.7.3.6 AFM cantilever calibrationAFMs are sensitive to very small forces in the piconewton range.
In order tomeasure these forces accurately, the stiffness of the probe must be191192C H A P T ER 7 : Scanning probe and particle beam microscopydetermined. Stiffness calibration procedures rely on either imposing knownforces on the probe, measuring the geometrical and material properties of theprobe, or measuring its thermal fluctuations.The cantilever’s spring constant is essentially dependent upon itscomposition and dimensions [33].
Nominal values listed by manufacturers may be incorrect by an order of magnitude and it is, therefore,necessary to determine the spring constant for each cantilever or for eachbatch of cantilevers from a wafer [34]. Parameters such as Young’smodulus (related to composition), and cantilever length and thickness,can be used in theoretical equations to calculate a spring constant [35].However, calculated values can be inaccurate due to the unknownmaterial properties of the cantilever (the stoichiometry of silicon nitride,for example, can vary from Si3N4 to Si5N4 [36]).
Furthermore, themeasurement of cantilever thickness, which is a dominant parameter intheoretical equations, is extremely difficult. The spring constant dependson the cantilever thickness to the third power, so even small uncertaintyin the thickness measurement will result in large variations in thecalculated spring constant [37].An accurate, but often destructive, way to measure spring constant is theadded-mass method [38]. In this method beads of known mass are attachedto the end of the cantilever. The additional mass causes the cantileverresonant frequency to decrease proportional to the mass.
A graph of addedmass against resonant frequency yields a straight line with a slope corresponding to the spring constant.A further method to determine the spring constant is the measurementof the force that an AFM imparts onto a surface by measuring the thermalfluctuations of the cantilever – in this method the cantilever is modelled asa simple harmonic oscillator (usually only in one degree of freedom) [39].With knowledge of the potential energy of the system and applying theequipartition theorem, the spring constant of the cantilever can be calculated from the motion of the cantilever and its surrounding heat-bathtemperature.
The thermal method has three major problems [40]:(a) higher vibration modes cannot be ignored, (b) the method to measuredeflection usually measures the inclination rather than the displacement,and (c) only the first modes are accessible due to the bandwidth limitationsof the experiments.For directly traceable measurements of the force an AFM cantileverimparts on a surface, electrostatic balances can be used, but they are verycostly and inconvenient (see section 10.3.3). Many of the devices discussedin section 10.3.4 can also be used to measure spring constant when used aspassive springs.Atomic force microscopy7.3.7 Inter- and intra-molecular force measurement using AFMAs discussed previously, the AFM images a sample by sensing andresponding to forces between a tip and the sample.
Because the force resolution of the AFM is so sensitive (0.1 pN to 1 pN), it is a powerful tool forprobing the inter- and intra-molecular forces between two substances.Researchers have taken advantage of this sensitivity to quantify fundamentalforces between a sample and some substance linked to the AFM cantilever ortip [41]. The AFM has enabled some truly remarkable advances in thephysical sciences due to the sensitivity and ranges of force it can measure.
Afew examples will be discussed here. A basic understanding of the forcesbetween the AFM tip and the sample is essential for a proper use of theinstrument and the analysis of the data. A variety of forces that come intoplay between the tip and the sample are summarized in Table 7.2. Thediscussion that follows will focus on contact-mode AFM, which is the mostcommonly used imaging mode.
A recent review highlights the effect ofsurface forces on dimensional measurements [30].The total force between the tip and the sample results from the sum ofvarious attractive and repulsive forces, as described below. As a model,consider the Lennard-Jones potential, which describes the change in intermolecular potential energy (f) that occurs as two particles, such as atoms ormolecules (on tip and sample), are brought closer together. The model gives s 12 s6(7.1)f ¼ 43rrwhere s is approximately the atomic or molecular diameter (distance ofclosest approach), 3 is the minimum value of the potential energy or thedepth of the potential energy well, and r is the separation distance [42].
Asthe particles are brought closer together from relatively distance separations,Table 7.2Examples of surface forces commonly encountered in AFM measurementType of forceDependence of energy on distance (d)Energy (kJ$mol1)Range (nm)Intra-molecular (ionic or covalent)London dispersionH-bondingDipolesElectrostaticVan der WaalsSolvationHydrophobic1/d1/d 61/d 31/d 3ed1/d~ed~ed100s1 to 315 to 205 to 1010 to 1001 to 51 to 101 to 5<10.5 to 50.5 to 30.5 to 310s to 100s5 to 10<510s to 100s193194C H A P T ER 7 : Scanning probe and particle beam microscopythe (1/r)6 term (i.e. Van der Waals term) describes the slow change inattractive forces.
As the particles are brought even closer together, the (1/r)12term describes the strong repulsion that occurs when the electron cloudsstrongly repel one another.The Van der Waals interaction forces are long-range, relatively weakattractive forces. The origin of the Van der Waals forces is quantummechanical in nature; they result from a variety of interactions, primarilyinduced dipole and quadrupole interactions. The Van der Waals forces arenonlocalized, meaning that they are spread out over many atoms. Van derWaals forces for a typical AFM have been estimated to be of the order of 10nN to 20 nN [43].The so-called atomic force (a result of the Pauli exclusion principle) is theprimary repulsive force at close approach.
The magnitude of this force isdifficult to predict without a detailed understanding of surface structure.Several additional forces or interactions must be considered for an AFMtip and sample surface. Capillary adhesion is an important attractive forceduring imaging in air. The capillary force results from the formation ofa meniscus made up of water and organic contaminants adsorbed on to thesurface of the tip and the sample [36] (see Figure 7.7). The capillary force hasbeen estimated to be of the order of 100 nN or greater. When the tip and thesample are completely immersed in liquid, a meniscus does not form and thecapillary forces are absent. Some tips and samples may have hydrophobicproperties, in which case hydrophobic interactions must also be taken intoconsideration.Water near hydrophilic surfaces is structured [34].
When the tip and thesample are brought into close contact during force microscopy in solution orhumid air, repulsion arises as the structured water molecules on the surfacesof the tip and the sample are pushed away. In aqueous solutions, electricaldouble-layer forces, which may be either attractive or repulsive, are presentFIGURE 7.7 Schematic illustration of the strong capillary force that tends to drive thetip and sample together during imaging in air.Atomic force microscopynear the surfaces of the tip and the sample. These double-layer forces arisebecause surfaces in aqueous solution are generally charged.Lateral frictional forces must also be taken into account as the sample isscanned beneath the tip. At low forces, a linear relationship should holdbetween the lateral force and the force normal (vertical) to the surface witha proportionality constant equal to the coefficient of friction.
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