JOURNAL OF APPLIED PHYSICS 109, 113528 (2011) Relationship between Q-factor and sample damping for contact resonance atomic force microscope measurement of viscoelastic properties P. A. Yuya,1 D. C. Hurley,2 and J. A. Turner1,a) 1Department of Engineering Mechanics, University of Nebraska-Lincoln W317.4 Nebraska Hall, Lincoln, Nebraska 68588-0526, USA 2Materials Reliability Division, National Institute of Standards and Technology, Boulder, Colorado 80305, USA (Received 15 February 2011; accepted 18 April 2011; published online 8 June 2011) Contact resonance AFM characterization techniques rely on the dynamics of the cantilever as it vibrates while in contact with the sample. In this article, the dependence of the quality factor of the vibration modes on the sample properties is shown to be a complex combination of beam and sample properties as well as the applied static tip force. Here the tip-sample interaction is represented as a linear spring and viscous dashpot as a model for sample (or contact) stiffness and damping. It is shown that the quality factor alone cannot be used to infer the damping directly. Experimental results for polystyrene and polypropylene are found to be in good agreement with predictions from the model developed. These results form the basis for mapping viscoelastic properties with nanoscale resolution. VC 2011 American Institute of Physics. [doi:10.1063/1.3592966] I. INTRODUCTION elastic properties of a sample. The quality factor Q for the contact modes of the beam is not proportional to the sample Although the atomic force microscope (AFM)1 was damping and must be determined with respect to its relation originally developed for topography imaging, it has more with the complex wave numbers of the cantilever. This theo- recently been exploited for characterization of mechanical retical approach leads to a method for mapping simultane- properties. Impressive quantitative and numerous qualitative ously the storage and loss moduli of a sample with CR-FM if results have been reported with various modifications of the contact resonance and Q values can be simultaneously the basic AFM system.2–9 One such modification, contact measured as the tip scans across the sample surface. Such resonance force microscopy (CR-FM)10–12 has become an measurements are now possible with a variety of meth- increasingly important technique for characterizing mechani- ods16,18,19 similar to those for imaging elastic properties cal properties of materials at submicrometer scales. CR-FM alone.20 methods use the resonant modes of the AFM cantilever in order to evaluate near-surface mechanical properties.2,3,5 The initial CR-FM work used the resonant frequencies alone, II. THEORY without consideration of the complete spectral response (i.e., peak width), in order to quantify local elastic properties. An AFM cantilever beam (with modulus E, density q Recently,13 the complete resonance was fit in order to quan- cross-sectional area A, bending moment of inertia I, length tify the viscoelastic response of the material. That work L) in contact with a viscoelastic surface is modeled as shown introduced the possibility of quantitative mapping of storage in Fig. 1 where q(x, t) is the displacement of the point x at and loss moduli with AFM spatial resolution. The connection time t. For simplicity, the tip-sample interaction is modeled between the material storage and loss moduli and the reso- as a Kelvin–Voigt element, which is assumed to represent nant frequency and the peak width is often used in dynamic the response of the material alone (rather than the contact). nanoindentation work to quantify viscoelastic properties.14,15 More complex tip-sample interaction models may be needed However, the dynamical system of the instrument is much for specific applications, but this simplified case allows our simpler for nanoindentation because it is often designed to main points to be clearly made with respect to a large class possess only one degree of freedom. Quantitative mapping of materials. In the case of a harmonic displacement, the of viscoelastic properties by AFM is still a challenge because complex force at the position of the tip is given by the properties of the sample are convolved with the dynamics Pcomplex ¼ (k þ ixc)q(L1, t), where L1 is the position of the of the AFM cantilever beam.3,13,16,17 Therefore quantitative tip from the clamped end, k is the real part and xc is the mapping of the viscoelastic properties of the sample is imaginary part of the contact stiffness. The flexural motion only possible if the vibrational dynamics of the cantilever- of the cantilever is often modeled with the Euler–Bernoulli tip-sample combination are well understood. beam equation for which the solution in relation to specific 3,13,17 In this article, we highlight the importance of the canti- CR-FM applications can be found elsewhere. Here we lever beam dynamics with respect to extraction of the visco- focus on the response of the beam to a harmonic force exci- tation (magnitude FD; frequency x) applied at the tip. The a)Author to whom correspondence should be addressed. Electronic mail: displacement of the beam can be expressed as a modal [email protected]. expansion of the form 0021-8979/2011/109(11)/113528/5/$30.00 109, 113528-1 VC 2011 American Institute of Physics Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 113528-2 Yuya, Hurley, and Turner J. Appl. Phys. 109, 113528 (2011) where the modal amplitude An ¼ FDYn(L1)Yn(x0)/mb depends on the mode shape at the positions of the tip and laser. Equa- tion (3) clearly has the typical damped Lorentzian form as expected. Next, the relation between the modal frequency and damping is found through the wavenumber. The dimen- sionless complex wavenumbers are defined such that cn ¼ an þ ibn, where an and bn are real constants. This form is substituted into Eq. (3) and rearranged, taking in consider- ation that bn an, for the case of small damping. Equation (3) then becomes FIG. 1. Mechanical model of an AFM cantilever tip in contact with a visco- elastic surface. The interaction at the AFM tip is approximated by a linear A eixt spring-dashpot system that represents the viscoelastic response of the sample. qx; t ÀÁn ÀÁ: (4) ðÞ0 4 2 3 Nan À x þ i xv þ 4Nanbn ixt X1 The resonant frequency of the nth mode is determined from FDe YnðÞL1 YnðÞx qxðÞ¼;t ÂÃðn ¼ 1;2;3…1Þ: (1) the real part of the complex wavenumber an of that particular m 4 2 b n¼1 Ncn À x þ ixv mode and the beam damping is increased from v (for the free 3 beam) by an amount 4Nanbn=xn (note the dependence on Here, n are the mode numbers, mb ¼ qAL is mass of the 3 an). At this stage, it is convenient to define the dimensionless beam, N ¼ EI/mbL and v ¼ v=qA are dimensionless beam frequency response function, G(ix)as constants (where v represents the assumed viscous damping in the beam), and c ¼ k L are the normalized wavenumbers. 1 n n GiðÞx ¼ ÀÁÀÁ; (5) The mode shapes (spatial eigenfunctions) are defined in gen- 2 2 2 2 1 À x =xn þ i xv þ 4xnbn=an =xn eral form as. YnðxÞ¼f ½sinðknLÞsinhðÞknL =½cosðÞknL 2 4 þ coshðknL Þg½þsinðÞÀknx sinhðÞknx ½cosðÞÀknx coshðÞknx where xn ¼ Nan is the natural frequency. The quality factor with the shape determined by the wavenumber kn. The final Q is defined as the maximum of the magnitude of Eq. (5), aspect of the beam model arises when the boundary condi- jjGiðÞx max. For small damping, this peak occurs when x tions (at x ¼ 0, L) and continuity conditions (at x ¼ L1) are xn such that Q for the nth mode is written as enforced. These operations result in the characteristic equa- x a tion that governs the wavenumbers. For the system shown in Q Gix Gix n n : (6) n ¼ jjðÞmax¼ jjðÞx¼xn ¼ Fig. 1, this equation reduces to ðÞanv þ 4xnbn Thus we see that for a given mode, QÀ1 can be decomposed 2 3 ðÞknL1 ½1 þ cos knL cosh knL into two parts: 3 h 0 0 ÀÁÀÁ 2 À1 À1 ¼ a þ ibkðÞnL1 1 þ cos knL cosh knL À1 free sample ðÞQn ¼ Qn þ Qn ; Â ðÞsinh k L cos k L À sin k L cosh k L ÀÁ n 1 n 1 n 1 n 1 free À1 where Qn ¼ v=xnÀÁis associated with the damping of the 0 0 sample À1 þðÞ1 À cos knL1 cosh knL1 sin knL cosh knL cantilever alone and Q ¼ 4 bn/an quantifies the i n 0 0 damping from the sample. AlthoughÀÁ such a decomposition À1 À cos knL sinh knL Þ ; (2) sample may be expected, the form for Qn is not intuitive because it is related directly to the wavenumber rather than wherepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia ¼ k/kc is the dimensionless stiffness and directly to the sample damping. This result is a consequence 2 b ¼ c L1=ðÞ9EIqA is the dimensionless damping, both of of the boundary value problem solved—the damping of a which are attributed here to the sample (although it is recog- specific mode arises from a local interaction that depends on nized that stiffness and damping may arise in the contact the amplitude of the eigenfunction (mode shape) at the tip itself due to effects such as adhesion).21,22 Equations (1) and position. Thus, the modal damping is a complex convolution (2) form the basis for the discussion that follows.