An extended Butterworth Van Dyke model for quartz crystalmicrobalance applications in viscoelastic fluid media

摘要

An extended Butterworth-Van Dyke (EBVD) model withnfrequency-independent parameters for the characterization of a resonantncompound formed by a quartz crystal in contact with a finitenviscoelastic layer contacting a semi-infinite viscoelastic medium isnextracted by analysis of the lumped element model. The formulation ofnthe EBVD model is compared with the complete expression of thenelectrical admittance of the loaded quartz derived from the transmissionnline model (TLM). Relative deviations between them do not exceed 3%naround 1% bandwidth near resonance. An extended Martin and Granstaff'snmodel and an explicit expression for the frequency shift that supposesnan extension of Kanazawa's model for viscoelastic media are obtained. Annanalysis of the errors associated with the extraction of shearnparameters of the coating for different materials prove that, to obtainnan error less than 5% in the shear parameters determination, thenviscoelastic contribution, defined as the relative error in thenthickness computed from the frequency shift by Sauerbrey equation, mustnbe greater than a limit that depends on Q, which is defined as the rationof the shear storage modulus (G') to shear loss modulus (G"). In thenmaterials studied polymers in the transition range or in the rubberynstate with Q=1 and 10, the viscoelastic contribution must be higher thann15% and 50%, respectively, for a 5% limit error in the shear parametersnextraction. A criterion for a practical determination of the appropriatenviscoelastic regimes is indicated