# Heterogeneous Samples

• Diethelm Johannsmann
Chapter
Part of the Soft and Biological Matter book series (SOBIMA)

## Abstract

When the sample is structured in the plane of the resonator with a characteristic scale comparable to the wavelength of sound, analytical predictions of the displacement field and the frequency shift are difficult. Among the samples that are heterogeneous in this sense are nanobubbles, nanodroplets, nanoparticles, vesicles, and biological cells. In analyzing such samples, one can rely on common sense and empirical correlations. If one wants to go beyond those more qualitative pictures, one can calculate the area-averaged periodic stress at the resonator surface numerically. An example of a numerical method is discussed in detail. The finite element method (FEM) is employed to solve the incompressible Stokes problem and to predict the periodic interfacial stress. The frequency shift follows from the area-averaged stress and the SLA.

## Keywords

Frequency Shift Capillary Pressure Capillary Number Slip Length Stoke Flow
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Glossary

Variable

3PL

As an index: 3-Phase Line

Area of a droplet

bS

Ca

Capillary number

D

Dissipation factor (D = 2Γ/f r )

D

As an index: Droplet

f

Frequency

f0

Resonance frequency at the fundamental (f 0 = Z q /(2m q ) = Z q /(2ρ q d q ))

G

Shear modulus

h

k

Wavenumber

liq

As an index: liquid

n

Overtone order

LD

Perimeter of a droplet

p

Pressure

rD

r

Position (a vector)

R

Radius of a liposome (Fig. 12.12)

S

As an index: Surface

t

Time

T

Temperature

Tm

Melting temperature of a lipid membrane

u, u

Tangential displacement (when bold: a vector)

v, v, $${\hat{\text{v}}}$$, $${\hat{\mathbf{v}}}$$

Velocity

vis

As an index: viscous

x, y, z

Spatial coordinates, z: along the surface normal

$$\tilde{Z}_{liq}$$

Shear-wave impedance of a liquid (Z̃ liq  = (iωρ liq η liq )1/2)

$${\dot{\upgamma }}$$

Shear rate

γS

Surface energy

Γ

Imaginary part of a resonance frequency

δ

Penetration depth of a shear wave (Newtonian liquids: δ = (2η liq /(ρ liq ω))1/2)

δL

Loss angle

Δ

As a prefix: A shift induced by the presence of the sample

$${\upeta ,\tilde{\upeta }}$$

Viscosity

θ

Coverage

ρ

Density

σ

Tangential stress

τr

Emulsion time (A relaxation time of emulsions and droplets)

ω

Angular frequency

## References

1. 1.
Hayden, O., Lieberzeit, P.A., Blaas, D., Dickert, F.L.: Artificial antibodies for bioanalyte detection-sensing viruses and proteins. Adv. Funct. Mater. 16(10), 1269–1278 (2006)
2. 2.
Bingen, P., Wang, G., Steinmetz, N.F., Rodahl, M., Richter, R.P.: Solvation effects in the quartz crystal microbalance with dissipation monitoring response to biomolecular adsorption. A phenomenological approach. Anal. Chem. 80(23), 8880–8890 (2008)
3. 3.
Olsson, A.L.J., Quevedo, I.R., He, D., Basnet, M., Tufenkji, N.: Using the quartz crystal microbalance with dissipation monitoring to evaluate the size of nanoparticles deposited on surfaces. ACS Nano 7(9), 7833–7843 (2013)
4. 4.
Grest, G.S.: Interfacial sliding of polymer brushes: a molecular dynamics simulation. Phys. Rev. Lett. 76(26), 4979–4982 (1996)
5. 5.
Urbakh, M., Daikhin, L.: Roughness effect on the frequency of a quartz-crystal resonator in contact with a liquid. Phys. Rev. B 49(7), 4866–4870 (1994)
6. 6.
Adamczyk, Z., Siwek, B., Zembala, M., Belouschek, P.: Kinetics of localized adsorption of colloid particles. Adv. Colloid Interface Sci. 48, 151–280 (1994)
7. 7.
Kastl, K., Herrig, A., Luthgens, E., Janshoff, A., Steinem, C.: Scrutiny of annexin A1 mediated membrane—membrane interaction by means of a thickness shear mode resonator and computer simulations. Langmuir 20(17), 7246–7253 (2004)
8. 8.
Wriggers, P.: Computational Contact Mechanics. Springer, Heidelberg (2006)Google Scholar
9. 9.
Johannsmann, D., Reviakine, I., Rojas, E., Gallego, M.: Effect of sample heterogeneity on the interpretation of QCM(-D) data: comparison of combined quartz crystal microbalance/atomic force microscopy measurements with finite element method modeling. Anal. Chem. 80(23), 8891–8899 (2008)
10. 10.
Granick, S.: Ferritin—its properties and significance for iron metabolism. Chem. Rev. 38(3), 379–403 (1946)
11. 11.
Hook, F., Rodahl, M., Brzezinski, P., Kasemo, B.: Measurements using the quartz crystal microbalance technique of ferritin monolayers on methyl-thiolated gold: dependence of energy dissipation and saturation coverage on salt concentration. J. Colloid Interface Sci. 208(1), 63–67 (1998)
12. 12.
Johnson, C.A., Yuan, Y., Lenhoff, A.M.: Adsorbed layers of ferritin at solid and fluid interfaces studied by atomic force microscopy. J. Colloid Interface Sci. 223(2), 261–272 (2000)
13. 13.
Hemmersam, A.G., Rechendorff, K., Besenbacher, F., Kasemo, B., Sutherland, D.S.: pH-dependent adsorption and conformational change of ferritin studied on metal oxide surfaces by a combination of QCM-D and AFM. J. Phys. Chem. C 112(11), 4180–4186 (2008)
14. 14.
Johannsmann, D., Reviakine, I., Richter, R.P.: Dissipation in films of adsorbed nanospheres studied by quartz crystal microbalance (QCM). Anal. Chem. 81(19), 8167–8176 (2009)
15. 15.
Reviakine, I., Gallego, M., Johannsmann, D., Tellechea, E.: Adsorbed liposome deformation studied with quartz crystal microbalance. J. Chem. Phys. 136(8), 84702–84705 (2012)
16. 16.
Pomorska, A., Shchukin, D., Hammond, R., Cooper, M.A., Grundmeier, G., Johannsmann, D.: Positive frequency shifts observed upon adsorbing micron-sized solid objects to a quartz crystal microbalance from the liquid phase. Anal. Chem. 82(6), 2237–2242 (2010)
17. 17.
Johannsmann, D.: Viscoelastic, mechanical, and dielectric measurements on complex samples with the quartz crystal microbalance. Phys. Chem. Chem. Phys. 10(31), 4516–4534 (2008)
18. 18.
Tellechea, E., Johannsmann, D., Steinmetz, N.F., Richter, R.P., Reviakine, I.: Model-independent analysis of QCM data on colloidal particle adsorption. Langmuir 25(9), 5177–5184 (2009)
19. 19.
Finger, A., Johannsmann, D.: Hemispherical nanobubbles reduce interfacial slippage in simple liquids. Phys. Chem. Chem. Phys. 13(40), 18015–18022 (2011)
20. 20.
Hook, F., Ray, A., Norden, B., Kasemo, B.: Characterization of PNA and DNA immobilization and subsequent hybridization with DNA using acoustic-shear-wave attenuation measurements. Langmuir 17(26), 8305–8312 (2001)
21. 21.
Reviakine, I., Brisson, A.: Streptavidin 2D crystals on supported phospholipid bilayers: toward constructing anchored phospholipid bilayers. Langmuir 17(26), 8293–8299 (2001)
22. 22.
Stone, H.A.: Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid Mech. 26, 65–102 (1994)
23. 23.
Minale, M.: Models for the deformation of a single ellipsoidal drop: a review. Rheol. Acta 49(8), 789–806 (2010)
24. 24.
Hyväluoma, J., Kunert, C., Harting, J.: Simulations of slip flow on nanobubble-laden surfaces. J. Phys. Condens. Matter 23(18), 184106 (2011)
25. 25.
Steinberger, A., Cottin-Bizonne, C., Kleimann, P., Charlaix, E.: High friction on a bubble mattress. Nat. Mater. 6(9), 665–668 (2007)
26. 26.
Du, B.Y., Goubaidoulline, E., Johannsmann, D.: Effects of laterally heterogeneous slip on the resonance properties of quartz crystals immersed in liquids. Langmuir 20, 10617–10624 (2004)
27. 27.
Zhang, X.H.: Quartz crystal microbalance study of the interfacial nanobubbles. Phys. Chem. Chem. Phys. 10(45), 6842–6848 (2008)
28. 28.
Lou, S.T., Ouyang, Z.Q., Zhang, Y., Li, X.J., Hu, J., Li, M.Q., Yang, F.J.: Nanobubbles on solid surface imaged by atomic force microscopy. J. Vac. Sci. Technol. B 18(5), 2573–2575 (2000)
29. 29.
30. 30.
Richter, R.P., Berat, R., Brisson, A.R.: Formation of solid-supported lipid bilayers: an integrated view. Langmuir 22(8), 3497–3505 (2006)
31. 31.
Reviakine, I., Johannsmann, D., Richter, R.P.: Hearing what you cannot see and visualizing what you hear: interpreting quartz crystal microbalance data from solvated interfaces. Anal. Chem. 83(23), 8838–8848 (2011)
32. 32.
Tamm, L.K., McConnell, H.M.: Supported phospholipid-bilayers. Biophys. J. 47(1), 105–113 (1985)
33. 33.
Sackmann, E.: Supported membranes: scientific and practical applications. Science 271(5245), 43–48 (1996)
34. 34.
Keller, C.A., Kasemo, B.: Surface specific kinetics of lipid vesicle adsorption measured with a quartz crystal microbalance. Biophys. J. 75(3), 1397–1402 (1998)
35. 35.
Reimhult, E., Hook, F., Kasemo, B.: Vesicle adsorption on SiO2 and TiO2: dependence on vesicle size. J. Chem. Phys. 117(16), 7401–7404 (2002)
36. 36.
Reviakine, I., Rossetti, F.F., Morozov, A.N., Textor, M.: Investigating the properties of supported vesicular layers on titanium dioxide by quartz crystal microbalance with dissipation measurements. J. Chem. Phys. 122(20), 204711 (2002)
37. 37.
Reviakine, I., Brisson, A.: Formation of supported phospholipid bilayers from unilamellar vesicles investigated by atomic force microscopy. Langmuir 16(4), 1806–1815 (2000)
38. 38.
Richter, R., Mukhopadhyay, A., Brisson, A.: Pathways of lipid vesicle deposition on solid surfaces: a combined QCM-D and AFM study. Biophys. J. 85(5), 3035–3047 (2003)
39. 39.
Edvardsson, M., Svedhem, S., Wang, G., Richter, R., Rodahl, M., Kasemo, B.: QCM-D and reflectometry instrument: applications to supported lipid structures and their biomolecular interactions. Anal. Chem. 81(1), 349–361 (2009)
40. 40.
Seifert, U.: Adhesion of vesicles in 2 dimensions. Phys. Rev. A 43(12), 6803–6814 (1991)
41. 41.
Seifert, U.: Configurations of fluid membranes and vesicles. Adv. Phys. 46(1), 13–137 (1997)
42. 42.
Mornet, S., Lambert, O., Duguet, E., Brisson, A.: The formation of supported lipid bilayers on silica nanoparticles revealed by cryoelectron microscopy. Nano Lett. 5(2), 281–285 (2005)
43. 43.
Lee, C.H., Lin, W.C., Wang, J.P.: All-optical measurements of the bending rigidity of lipid-vesicle membranes across structural phase transitions. Phys. Rev. E 64(2), 020901 (2001)
44. 44.
Yi, Z., Nagao, M., Bossev, D.P.: Bending elasticity of saturated and monounsaturated phospholipid membranes studied by the neutron spin echo technique. J. Phys. Condens. Matter 21(15), 15510 (2009)
45. 45.
Heitmann, V., Reiss, B., Wegener, J.: The quartz crystal microbalance in cell biology: basics and applications. In Steinem, C., Janshoff, A. (eds.) Piezoelectric Sensors. Springer, Berlin (2007)Google Scholar
46. 46.
Sapper, A., Wegener, J., Janshoff, A.: Cell motility probed by noise analysis of thickness shear mode resonators. Anal. Chem. 78(14), 5184–5191 (2006)
47. 47.
Pax, M., Rieger, J., Eibl, R.H., Thielemann, C., Johannsmann, D.: Measurements of fast fluctuations of viscoelastic properties with the quartz crystal microbalance. Analyst 130(11), 1474–1477 (2005)
48. 48.
Li, J., Thielemann, C., Reuning, U., Johannsmann, D.: Monitoring of integrin-mediated adhesion of human ovarian cancer cells to model protein surfaces by quartz crystal resonators: evaluation in the impedance analysis mode. Biosens. Bioelectron. 20(7), 1333–1340 (2005)
49. 49.
Tessier, L., Patat, F., Schmitt, N., Lethiecq, M., Frangin, Y., Guilloteau, D.: Significance of mass and viscous loads discrimination for an at-quartz blood-group immunosensor. Sens. Actuators B-Chem. 19(1–3), 698–703 (1994)
50. 50.
Bandey, H.L., Cernosek, R.W., Lee, W.E., Ondrovic, L.E.: Blood rheological characterization using the thickness-shear mode resonator. Biosens. Bioelectron. 19(12), 1657–1665 (2004)
51. 51.
Muller, L., Sinn, S., Drechsel, H., Ziegler, C., Wendel, H.P., Northoff, H., Gehring, F.K.: Investigation of prothrombin time in human whole-blood samples with a quartz crystal biosensor. Anal. Chem. 82(2), 658–663 (2010)
52. 52.
Saitakis, M., Gizeli, E.: Acoustic sensors as a biophysical tool for probing cell attachment and cell/surface interactions. Cell. Mol. Life Sci. 69(3), 357–371 (2012)

© Springer International Publishing Switzerland 2015

## Authors and Affiliations

1. 1.Institute of Physical ChemistryClausthal University of TechnologyClausthal-ZellerfeldGermany