Areal mass density

Acoustic Wave Sensors. Another emerging physical transduction technique involves the use of acoustic waves to detect the accumulation of species in or on a chemically sensitive film. This technique originated with the use of quartz resonators excited into thickness-shear resonance to monitor vacuum deposition of metals (11). The device is operated in an oscillator configuration. Changes in resonant frequency are simply related to the areal mass density accumulated on the crystal face. These sensors, often referred to as quartz crystal microbalances (QCMs), have been coated with chemically sensitive films to produce gas and vapor detectors (12), and have been operated in solution as Hquid-phase microbalances (13). A dual QCM that has one smooth surface and one textured surface can be used to measure both the density and viscosity of many Hquids in real time (14). 

The randomly distributed sheet clearly exhibits areas of low and high density but it still remains a good target and a unique reference structure in making paper. The small scale non-uniformities of paper structure are particularly important in their influence on pore size distribution and the distribution of areal mass density, and both of these properties have an influence on mechanical and other properties of the final sheet. 

Ah = vs At will be devoid of particles, and a particle layer of areal mass density rsw A/z = rsw vs At (kgsm 2) covers the bottom. Thus, the initial relative particle removal rate ks is given by ... 

These correspond, respectively, to polymer or electrolyte entrapped within surface features, the polymer film, and the solution. The first of these is a minor effect when using polished crystals the surface mechanical impedance of this contribution is Z, = wp where j = V-l, o> = 2nf0, and p is the areal mass density of the entrapped material. For finite and semiinfinite viscoelastic layers, the surface mechanical impedance is given by Z, = (GPf)m and Zv = (Gpf)1/2 tanh(y/i/), respectively, where prf and hf are the film density and thickness and y = /w(p/G)l/2. For the solution, Zs = (tapsT]J2)m (1 + j), where p, and tj, are the density and viscosity of the solution. When rigid mass, finite viscoelastic film and semi-infinite liquid loadings are all present, as in the experiment of Fig. 13.7, one can show that [42] ... 

An ideal mass layer is assumed to have an infinitesimal thickness, yet contribute a finite areal mass density to the device surface. In Section 3.1.1, we noted that this criterion holds as long as the acoustic phase shift across the film is small compared with ir. The equivalent-circuit model for the mass-loaded resonator can be determined from the surface mechanical impedance contributed by a surface perturbation. The surface stress required to sinusoidally accelerate a mass layer is [14]... 

Chemically sorbent films are commonly coated on TSM resonators to construct gas or vapor sensors. The absorption of species by diese films leads to a change in the areal mass density as well as plasticization or softening of the film. Com-... 

In Section 3.2.4 we considered the effects of an ideal mass layer on SAW response. In the model used to derive the mass-loading response, the layer was assumed to be (1) infinitesimally thick, and (2) subject only to translational motion by the SAW. Translational motion was found to induce a change in SAW velocity proportional to the areal mass density (pfc) contributed by the film — the mass loading response. Since no power dissipation arises in film translation, no attenuation response was predicted. With an actual film having finite thickness and elastic properties, it is important to also consider the effects of SAW-induced film deformation. Energy storage and power dissipation due to film deformation cause additional contributions to SAW velocity and attenuation that were neglected in the earlier treatment. 

Figure 335 Fractional change in APM propagation velocity vs areal/mass density of silver deposited onto the device surface. (Reprinted with permission. See Ref. [54]. 1989 Elsevier Publishers.)...

Here and in the following, /o is the resonance frequency of the crystal in the reference state (which usually is the uncoated state) mf and mq are the areal mass densities (mass per unit area) of the film and the crystal, respectively. The relation df/dq = mil mq evidently requires that the density of the film and the crystal are the same. It will turn out that the fractional frequency shift is the same as the ratio of mf and mq for all thin films, regardless of fheir acoustic properties. Therefore, one may memorize the relation A///o =- mf/mq right here. Note that this Sauerbrey limit only holds for films much thinner than the wavelength of sound, L. 

Equation 52 shows that the areal mass density of the crystal is the only parameter connecting the load and the frequency shift, as long as the latter is small. The stiffness of the crystal (and piezoelectric stiffening, in particular) is of no influence at this level of approximation. Comparing Eqs. 51 and 39, we find ... 

Before going into the details of the calculation for thin films, we briefly come back to a statement made earlier with regard to the proportionality of frequency shift and added mass (as opposed to film thickness). This propor-tionaUty is the essence of the Sauerbrey relation. The frequency shift-mass proportionality holds for all thin films, regardless of their viscoelastic properties. It even applies to laterally heterogeneous samples as long as these are so thin that viscoelasticity can be ignored. In the latter case, the areal mass density of course is an average mass density. 

The correct interpretation of the frequency shift from QCM experiments in liquids is a challenge. Practitioners often just apply the Sauerbrey equation (Eq. 28) to their data and term the resulting areal mass density Sauerbrey mass and the corresponding thickness Sauerbrey thickness . Even though the Sauerbrey thickness can certainly serve to compare different experiments, it must not be naively identified with the geometric thickness. Here is a fist of considerations ... 

The QCM always measures an areal mass density, never a geometric thickness (cf. the remarks below Eq. 71). The conversion from areal mass density to thickness usually requires the physical density as an independent input. A density of 1 g cm is often assumed in soft matter experiments. Given the other uncertainties (see below), this is in many cases a fair approximation. 

If the spheres are small enough, they can be treated Hke a Sauerbrey film with an areal mass density mf = Nstns/A. Let /Cp be the spring constant of the crystal in the BvD sense, nip the equivalent mass of the crystal in BvD sense (Eq. 116 in Chap. 2 in this volume), and mq the areal mass density of the... 

For typical fluids (water and most organic solvents), xl (at 10 MHz) is on the order of 170 nm. The effectively coupled mass of this fluid layer, expressed as an areal mass density, is simply the product of Xl and the fluid density. For a smooth 10-MHz resonator, immersion in an aqueous solution couples ca. 17 ngcm of fluid, resulting in a decrease in the resonant frequency of ca. 3.8 kHz. Beyond this relatively small (cf 10 MHz operating frequency) baseline shift , the EQCM can then be used in situ to study interfadal processes under potential control. 

Figure 10. Dependence of serum adsorption on the extent of surface PEG side chain packing density. Z72i g is calculated as described in section 4.2. Values from this work are compared to published data, for which both the areal mass density of proteins and the average PEG spacing (L) are taken from refs 53 and 54. The radii of gyration were ealculated according to eq 1 for PEG, whereas the original value of 0.42 nm was used for the oligo(ethylene glyeol).

Berg S, Johannsmann D (2001) Laterally coupled quartz resonators. Anal Chan 73(6) 1140-1145 Stafford GR, Bertocci U (2009) In situ stress and nanogravimetric measurements during hydrogen adsorption/absorption on Pd ovo-layers deposited onto (lll)-Textured Au. J Phys Chem C 113 13249-13256 Way AS (1993) Quartz resonator techniques for simultaneous measurement of areal mass density, lateral stress, and temperature in thin films. Vacuum 44(3/4) 385-388... 

Studies on the woven fabrics using plain weaves made from high density polypropylene monofilaments were carried out using ASTM D150 standard method [46]. Four fabrics were used, each of which had different filament diameter, yam spacing, thickness and areal mass density. Each of these fabrics was kept between the capacitor plates and the corresponding volume fraction was calculated from the fabric areal density, density of the fiber and the electrode gap. The effective permittivity of the capacitor system with the fabric was calculated from the recorded capacitance value and the over-all capacitor dimensions using... 

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