Liquid in a tube oscillating along its axis

Author’s Accepted Manuscript Liquid in a tube oscillating along its axis Vladimir P. Zhdanov, Bengt Kasemo

www.elsevier.com/locate/physe

PII: DOI: Reference:

S1386-9477(15)00082-X http://dx.doi.org/10.1016/j.physe.2015.02.019 PHYSE11890

To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 19 February 2015 Accepted date: 20 February 2015 Cite this article as: Vladimir P. Zhdanov and Bengt Kasemo, Liquid in a tube oscillating along its axis, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2015.02.019 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Liquid in a tube oscillating along its axis Vladimir P. Zhdanova,b,∗ , Bengt Kasemoa a

Department of Applied Physics, Chalmers University of Technology,

S-412 96 G¨ oteborg, Sweden b

Boreskov Institute of Catalysis, Russian Academy of Sciences,

Novosibirsk 630090, Russia Abstract The Quartz Crystal Microbalance with Dissipation (QCM-D) sensing technique has become widely used to study various supported thin films and adsorption of biological macromolecules, nanparticles, aggregates, and cells. Such sensing, based on tracking shear oscillations of a piezoelectric crystal, can be employed in situations which are far beyond conventional ones. For example, one can deposit tubes on the surface of a sensor, orient them along the direction of the sensor surface oscillations, and study liquid oscillations inside the oscillating tubes. Herein, we illustrate and classify theoretically the regimes of liquid oscillations in this case. In particular, we identify and scrutinize the transition from the regime with appreciable gradients along the radial coordinate, which are qualitatively similar to those near the oscillating flat interface, to the regime where the liquid oscillates nearly coherently in the whole tube. The results are not only of relevance for the specific case of nanotubes but also for studies of certain mesoporous samples. Keywords: Fluid mechanics; Liquid oscillations; Shear force; Energy dissipation; Mesoporous solids; Nanotubes; Quartz Crystal Microbalance with Dissipation. ∗

E-mail address: [email protected]

1

Oscillations at a liquid-solid interface induce oscillations in the liquid. An interesting special case here are shear oscillations when the interface oscillates in its plane. In this case, due to viscosity, the oscillations in the liquid are successively more and more damped damp with increasing distance from the interface. If (i) the interface is flat and oscillates harmonically, (ii) the liquid compression is negligible, and (iii) the thickness of the liquid layer is much larger than the penetration depth of the perturbation, the established oscillations of liquid are well know to be described analytically [1]. In particular, the liquid velocity, v, is oriented along the interface, depends only on the coordinate, x ≥ 0, perpendicular to the interface (x = 0 corresponds to the interface), and satisfies the following equation ∂2v ∂v = ν 2, ∂t ∂x

(1)

where ν is the coefficient of kinematic viscosity. The solution of this equation (Fig. 1) is v = v0 exp(−x/δ) cos(x/δ − ωt),

(2)

where ω is the frequency of oscillations, δ = (2ν/ω)1/2 is the penetration depth, and v0 is the maximal interface velocity. The force acting per unit interface area is accordingly given by 

∂v   F=η = −ρ(ων)1/2 v0 cos(ωt + π/4),  ∂x x=0

(3)

where ρ is the liquid density, and η = ρν is the coefficient of dynamic viscosity. These equations show that due to the interplay of liquid viscosity and inertia there is an increasing ”phase-lag” of the oscillations in the liquid with increasing distance from the surface (note e.g. that the v = 0 condition moves from left to right as δ increases). In reality, the liquid oscillations described above may occur in various situations. In particular, such oscillations represent one of the ingredients of the function of the QCM-D sensing systems [2, 3, 4, 5] which have become widely used to study various 2

thin films and adsorption of biological macromolecules (e.g., proteins, DNA, polymers, and polyelectrolytes), nanopartices, aggregates (e.g., vesicles or virions), and cells. In this setup, the typical frequency and penetration depth are 5 MHz and (in water) 250 nm, respectively (for other liquids, δ may be smaller or larger). In principle, the QCMD sensors can be employed in the situations which are far beyond the conventional ones. For example, this technique was employed in situ to quantify, in real time, adsorption of dye and coadsorbates on flat and mesoporous TiO2 films [6]. Following this line, one can in principle deposit tubes on the surface of a sensor and orient them along the direction of the surface oscillations. In this context (and perhaps in some other contexts as well), it is thus of interest to clarify how liquid will oscillate in a tube oscillating along its axis. This is a main goal of our communication. Concerning the subject under consideration, we may note that oscillations of liquid in a tube were earlier analyzed in different contexts (see, e.g., articles [7]-[16] and references therein). Usually, a tube was considered to be fixed. In particular, Womersley [7] described the liquid oscillations induced by oscillations of the pressure gradient. His results have been used directly or with extensions in many subsequent experimental and theoretical studies (see, e.g., [8, 9] and references therein). Some other related situations were described as well [10]-[16]. If a tube oscillates along its axis, the liquid located inside oscillates along this axis as well. Assuming the tube cross section is circular, we can use the polar coordinate, 0 ≤ r ≤ R (R is the tube radius), in order to describe liquid oscillations. In particular, Eq. (1) can be replaced by ν ∂ ∂v ∂v = r . ∂t r ∂r ∂r

(4)

Near the tube wall, the liquid velocity is equal to the wall velocity, i.e., v(R) = v0 cos(ωt).

(5)

The second boundary condition can be obtained taking into account that due to the 3

symmetry there is no force in the center, and accordingly 

∂v   = 0. ∂r r=0

(6)

Eq. (4) can formally be solved by using the zeroth-order Bessel functions. With the boundary conditions (5) and (6), the corresponding solution is, however, cumbersome, and to illustrate the final results one should perform numerical calculations. Under such circumstances, direct numerical integration of Eq. (4) appears to be more convenient. Following the latter way, we used dimensionless variables, = r/R and τ = ωt. With these variables, Eqs. (4)-(6) are read as A ∂ ∂v ∂v = , ∂τ ∂ ∂

(7) 

∂v   v(R) = v0 cos(τ ), and = 0, ∂ =0

(8)

where A = ν/ωR2 is the dimensionless parameter determining qualitative features of the liquid oscillations. The integration was performed by employing the conventional discrete scheme at 0.05 ≤ A ≤ 5 with Δτ = 10−6 and Δ = 0.01. With these steps, the integration was proved to be accurate. The results of our calculations (Figs. 2-4) show that with increasing A there is transition from the regime with appreciable gradients along r which are qualitatively similar to those described by Eqs. (1) and (2) (cf. Fig. 1 with Fig. 2 for A = 0.05; note that r/R = 1 is at the tube-liquid interface and should be compared with x/δ = 0 in Fig. 1) to the regime where the liquid oscillates nearly coherently in the whole tube (see, e.g., Fig. 4 for A = 5). To characterize the regimes of liquid oscillations, it is instructive to compare the maximal liquid-induced shear force, ρν(∂v/∂r), acting per unit area of the wall in a tube with that, ρ(ων)1/2 v0 , acting on the flat wall (Eq. (3)). The corresponding dimensionless parameter defined as the ratio of these forces is 

ν p1 = ω

1/2





1 ∂v  A1/2 ∂v  ≡ ≡ A1/2 D,   v0 ∂r r=R v0 ∂ =1 4

(9)

where D ≡ (∂v/∂ρ)/v0 is the normalized velocity gradient near the wall (at ρ = 1). Another relevant dimensionless parameter can be obtained by dividing the maximal force, ρν(∂v/∂r), acting per unit area of the wall by that, 0.5ρωRv0 , needed to induce coherent liquid oscillations, 



2ν ∂v  2A ∂v    ≡ ≡ 2AD. p2 = ωRv0 ∂r r=R v0 ∂ =1

(10)

The dependence of these parameters on A is shown in Fig. 5. As one could expect, p1 is lower than 1, because the amount of liquid in a tube per unit interface area is limited. As a function of A, p1 reaches a maximum and becomes close to 1 at A  0.16. In this case, the liquid oscillations in a tube are close to those near the flat wall, because the ratio of the penetration depth (Eq (2)) and the tube radius, δ/R ≡ (2ν/ωR2 )1/2 ≡ (2A)1/2 , is close to 1. With increasing A, as one could also expect, p2 increases and eventually becomes close to 1. In the context of QCM-D sensing, it is of interest to calculate the shift of the QCM frequency, Δf , due to oscillation of liquid in the tubes attached to the QCM surface (Δf measures the dynamic mass load). For the flat QCM surface, the frequency shift related to liquid is known to be described as [4, 17] Δf 1 = f ρq lq



ρη 2ω

1/2

,

(11)

where f ≡ ω/2π is the frequency in the vacuum, and ρq and lq are the density and thickness of the QCM plate. In the case of tubes, the force is changed (compared to that corresponding to the flat surface; Eq. (3) by a factor of 2πRNp1 , where N is the number of tubes per unit length (in the direction perpendicular to the tube axis), 2πRN is the tube wall area per unit area of the QCM surface, and p1 is the dimensionless factor defined by Eq. (9). This factor determines also the change of the frequency compared to that corresponding to the flat surface (Eq. (11)). Thus, the shift can be represented as 2πRNp1 Δf = f ρq lq



ρη 2ω

1/2

.

(12) 5

For a relatively thin close-packed layer of tubes, we have 2RN  1. Thus, the dimensionless factor determining the value of the shift, 2πRNp1 , can be close to πp1 . The parameters p1 and p2 , defined by expressions (3) and (3), and expression (12) for the QCM frequency shift characterize the shear force acting at the wall in a tube. The amount of dissipated energy (the D factor in QCM-D) can be numerically calculated in a similar way using the standard prescriptions of fluid mechanics [1]. Now, it is appropriate to make a few comments on the experimental opportunities to make real QCM or QCM-D experiments related to the treatment above. Firstly, there exists a manyfold of axial nanotube structures of different diameters that can apparently be used for these experiments (see, e.g., [18] and references therein). Such nanotubes can be deposited as a close packed monolayer with their axis oriented along the direction of shear oscillation, as assumed in deriving Eq. (12), or as multilayers of parallel tubes. In both cases, there will be a contribution from the infinite liquid outside the layer of nanotubes, that need to be subtracted by control experiments and/or calculations of the same type as shown here. Multilayers offer an advantage in this case, since the contribution from the tubes is then relatively much larger than in the case of a monolayer. However, there is then a challenge to fill the space outside/between the tubes so that liquid cannot oscillate there too (otherwise, one must include that liquid too in the theoretical treatment). For sufficiently small diameter tubes, the need of such subtractions maybe eliminated by using capillary condensation in the tubes in the vapor phase. The other challenges and difficulties here are (i) to make mono disperse sets of these nanotubes (same diameter, same length several mm), and finally the most difficult step (ii) to place a mono or multilayer of such tubes lying parallel to each other on the QCM sensor surface. This is admittedly a difficult experiment and not the one to start with, since there is a much easier way as follows. The straightforward way is to use electron beam lithography to etch parallel channels of suitable width and depth in a film material deposited on the sensor surface

6

and in the direction that coincides with the oscillation direction of the sensor. For this approach almost any material that can be deposited as a film could be employed. We would suggest to use either a polymer like PMMA or silicon (the channel surfaces would be SiO2 in this case). The last step in the preparation would be to place a lid on top of the structures. This could possibly be done by placing a very thin microscope cover glass on top as described in [19]. However, such a cover glass may still be too thick. In this case, a very thin film of ca. 1000 nm would be deposited on a dissolvable substrate and then floated off in the dissolving liquid, and finally picked up on the surface of the sample with the nano fabricated grooves. This is a well known procedure used in preparation work of thin films for electron microscopy. The parallel tubes manufactured in this case would not have perfect circular cross section without additional treatment. However, they would have rounded corners, and the roundness could be increased by various nano fabrication tricks such as under-etching. If a polymer is used rounding off could also be made by heating. Generally the approach outlined above is in our opinion very feasible and not of unusual experimental difficulty. Therefore the theory we present can be verified. When a set of samples are available for this type of experiments, one also may take advantage of the opportunity to test the model using liquids with different viscosities. Experimentally, one can furthermore employ the fact that the measurements are possible at different odd overtones, 15, 25, etc. MHz with different penetration depths in the liquid. Finally, we may add that our results are basic and generic and accordingly are expected to be used in one way or another in various areas beyond that declared in the beginning of our presentation and discussed above.

7

References [1] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1989, §24. [2] M. Rodahl, F. H¨o¨ok, A. Krozer, P. Brzezinski, B. Kasemo, Rev. Sci. Instr. 66 (1995) 3924. [3] M. Rodahl, F. H¨oo¨k, C. Fredriksson, C.A. Keller, A. Krozer, P. Brzezinski, M. Voinova, B. Kasemo, Faraday Disc. 107 (1997) 229. [4] M.V. Voinova, M. Rodahl, M. Jonson, B. Kasemo, Phys. Scripta 59 (1999) 391. [5] K.A. Marx, Biomacromol. 4 (2003) 1099. [6] H.A. Harms, N. Tetreault, V. Gusak, B. Kasemo, M. Gr¨atzel, Phys. Chem. Chem. Phys. 14 (2012) 9037. [7] J.R. Womersley, J. Physiol. 127 (I955) 553. [8] C. Blake, J. Edmunds, L. Shelford, J. Moger, S.J. Matcher, Proc. SPIE 6847 (2008) 684720. [9] N.E. Daidzic, J. Fluid. Eng. 136 (2014) 041102. [10] N.S. Deshpande, M. Barigou, Chem. Eng. Sci. 56 (2001) 3845. [11] G. Pontrelli, Med. Biol. Eng. Comput. 40 (2002) 550. [12] D.-Y. Shin, P. Grassia, B. Derby, Trans. ASME 127 (2005) 98. [13] S.P. Das , V.S. Nikolayev, F. Lefevre, B. Pottier, S. Khandekar, J. Bonjour, Intern. J. Heat Mass Transf. 53 (2010) 3905. [14] D. Yin, H.B. Ma, Intern. J. Heat Mass Transf. 66 (2013) 699. [15] R. Drelich, M. Pakula, M. Kaczmarek, Transp. Por. Med. 101 (2014) 69. 8

[16] M. Karbaschi, A. Javadi, D. Bastani, R. Miller, Coll. Surf. A: Physicochem. Eng. Asp. 460 (2014) 355. [17] C.E. Reed, K. K. Kanazawa, J.H. Kaufman, J. Appl. Phys. 68 (1990) 1993. [18] Y.N. Xia, P.D. Yang, Y.G. Sun, Y.Y. Wu, B. Mayers, B. Gates, Y.D. Yin, F. Kim, Y.Q. Yan, Adv. Mater. 15 (2003) 353. [19] G. Ohlsson, C. Langhammer, I. Zoric, B. Kasemo, Rev. Sci. Instrum. 80 (2009) 083905.

9

Figure captions Fig. 1. Shear liquid oscillations near the flat wall harmonically oscillating along its plane: (a) the normalized liquid velocity, v/v0 , as a function of time during one oscillation period (0 ≤ ωt ≤ 2π) for x/δ = 0, 1, 2, and 3 (Eq. (2)); (b) as (a) for the normalized force, F = F /[ρ(ων)1/2 ], acting per unit area of the wall (Eq. (3)); (c) the normalized liquid velocity as a function of the coordinate perpendicular to the wall for ωt = 0, π/2, π, and 3π/2 (Eq. (2)). Fig. 2. Shear liquid oscillations in a tube at A = 0.05: (a) the normalized velocity, v/v0 , as a function of time during one oscillation period (0 ≤ ωt ≤ 2π) for r/R = 0, 0.25, 0.5, 0.75 and 1; (b) the normalized velocity gradient, D ≡ (∂v/∂ )/v0 , near the wall (at r/R = 1) as a function of time; and (c) the normalized velocity as a function of r/R for ωt = 0, π/2, π, and 3π/2. Fig. 3. As Fig. 2 for A = 0.5. Fig. 4. As Fig. 3 for A = 5. Fig. 5. Normalized liquid velocity gradient, D ≡ (∂v/∂ )/v0 , near the tube wall (at r/R = 1) and parameters p1 ≡ A1/2 D and p2 ≡ 2AD as a function of A (A ≡ ν/ωR2 ).

10

(a)

1.0 0

0.5

v / v0

2

3

0.0

1

-0.5 -1.0

(b)

1

F

0

-1 0

2

ωt

4

6

(c)

v / v0

1.0

0.5

0.0

-0.5

-1.0

0

0

π 1

π/2

3π/2

x/δ

2

3

(a)

1.0

A = 0.05

v / v0

0.5 0

0.0 1

-0.5

0.75

(b)

D

2 0 -2 -4

0

2

ωt

4

6

(c)

v / v0

1.0

0.5

0.0

-0.5

-1.0 0.0

A = 0.05

0.2

0.4

r/R

0.6

0

π/2

π

3π/2

0.8

1.0

(a)

1.0

A = 0.5

v / v0

0.5 0.0

0 1

-0.5

(b) 0.5

D

0.0 -0.5 -1.0 0

2

ωt

4

6

(c)

v / v0

1.0

0.5

0.0

-0.5

-1.0 0.0

A = 0.5

0.2

π/2 3π/2

0.4

r/R

0.6

0

π

0.8

1.0

(a)

1.0

A=5

v / v0

0.5 0.0 -0.5

(b) 0.05

D

0.00 -0.05 -0.10

0

2

ωt

4

6

(c)

v / v0

1.0

0.5

0.0

-0.5

-1.0 0.0

A =5

0.2

π/2 3π/2

0.4

r/R

0.6

0.8

0

π

1.0

8

D

6 4 2 0

p1

1.0

0.5

0.0

p2

1.0

0.5

0.0

-2

-1

log10( A )

0

1