PHYSICAL REVIE%' 8 VOLUME 34, NUMBER 2 15 JULY 1986 Q factors of tluartx oscillator modes as a probe of submonolayer-film dynamics A. Widom and J. Krim Physics Department, Northeastern University, Boston, Massachusetts 02115 (Received 4 December 1985; revised manuscript received 21 April 1986) That frequency shifts by quartz crystals oscillating in the transverse shear mode can be used to determine adsorption isotherms for monolayer and submonolayer adsorbed films is dwell under- stood. Here, the nature of the dynamical information contained in the quality factors of such modes is theoretically discussed. It is well established that the frequency shift of a quartz crystal oscillating in the transverse shear mode (Fig. 1) provides a measure of the quantity of adsorbed mass on its surface. ' Adsorption isotherms in the submonolayer re- gime can be measured by such quartz-crystal microbal- ances, the theoretical basis of which is quite simple. If Mo denotes the mass of the quartz oscillator, and M denotes the mass of the adsorbed film, then Moot@ ~M0+ M represents the total effective mass of the mode. In the regime (2) Eq. (I) yields a small but measurable frequency shift uniform substrates. Such systems provide ideal candi- dates for studies of two-dimensional phase transitions. The transverse shear mode has been treated for oscilla- tions occurring in a thick viscous medium of finites and in- finite9 extent. Here the viscous interaction in the fluid ex- erts a tangential damping force whose real and imaginary parts are proportional to the increase in virtual inertia (or effective mass load) of the electrode and to the increase in dissipation, b(1/Qo). The theory of quartz-oscillator line shapes with adsorbed submonolayer films has not (to our knowledge) been worked out iii detail. We consider here the frictional damping force due to random fluctuations in adsorption forces. The central result of our considerations is that the quali- ty factor Q of the mode obeys 0 Mo from which the adsorbed mass can be deduced. 2 In Eq. (3), roo represents the resonant frequency of the mode. (bro/roo for an adsobed monolayer on an AT-cut quartz crystal is typically 10 -10, typical Q factors ranging from 10s-10r). The topic of high- oscillators is relevant to a variety of fields within physics, including the measurement of physi- cal properties of adsorbed films. Q factors of such oscilla- tors have been employed to measure a variety of properties ranging from superfluid mass and dissipation in two- and three-dimensional helium films, to the viscosity of buta- diene rubber films during oxidation. Moreover, the sensi- tivity of the Q factor to monolayer-thick films is well es- tablished. 6 Recent breakthroughs have allowed us, for the first time, to employ a high-Q resonator (here, an AT-cut quartz crystal) for gas-adsorption studies on atomically where Q and Qo are the quality factors with and without the adsorbed layer, and r, represents a dynamic correla- tion time in the film. The discussion which follows makes this notion precise by relating r, to the momentum fluc- tuations in the film. For simplicity of presentation, classical correlations will be used. The fully quantum mechanical theory is quite direct, and will be pursued in future work. From a physical viewpoint, the dissipative processes in the film give rise to a dynamical damping coefficient y(g), and thus a line-shape width contribution roob Rey(coo+i 0 ), 1 ~ + 2Qo where the approximation stated in Eq. (2) has been em- ployed. From a correlation function viewpoint, the fric- tional damping of the oscillator mode is caused by random fluctuations in adsorption forces. Thus, the dynamical frictional damping coefficient is given by the Kubo for- mula" ks TMoy(g) dt (F„(r)F„(0)) exp(i t, "r ), FIG. 1. Transverse shear mode of oscillation. ~here F„denotes the random force exerted on the quartz- oscillator coordinates in the x direction by the adsorbed film. Since the random forces exist by virtue of the film momentum fluctuations AP„, it is of use to consider the 34 1403 1986 The American Physical Society