Impact of Compressibility on the Control of Bubble-Pressure

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Impact of Compressibility on the Control of Bubble-Pressure Tensiometers Vineeth Chandran Suja, John M. Frostad, and Gerald G. Fuller Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b03258 • Publication Date (Web): 31 Oct 2016 Downloaded from http://pubs.acs.org on November 6, 2016

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Impact of Compressibility on the Control of Bubble-Pressure Tensiometers V. Chandran Suja ,† J. M. Frostad,†,‡ and G. G. Fuller∗,† Department of Chemical Engineering, Stanford University, Stanford, California 94305 E-mail: [email protected]

Abstract An experimental and theoretical investigation is conducted to understand the role of compressibility on the quasi-static expansion and contraction of a bubble that is pinned at the opening of a small capillary. The results show that there are two regimes of expansion and contraction depending on the values of two dimensionless parameters which correspond to a dimensionless volume and maximum capillary pressure. In one regime, not all bubble sizes are accessible during expansion and contraction, and the bubbles exhibit a hysteretic behaviour when cycling through expansion and contraction. We call this the bubble shape hysteresis. The magnitude of the bubble shape hysteresis is computed for a realistic range of the non dimensional parameters. In the other regime, the bubble size can be varied continuously, but compressibility can still make it difficult to smoothly control the size of the bubble. The theoretical analysis shows that compressibility affects the evolution of the bubbles, even when the bubble is smaller than a hemispherical cap. The analysis also provides the infusion and withdrawal rates that a syringe pump must supply to expand and contract the bubble at a desired rate, accounting for compressibility. The validity of the assumptions used in the model are verified by comparison against experimental data. ∗

To whom correspondence should be addressed Stanford University ‡ Present Address: Chemical and Biological Engineering, University of British Columbia, Vancouver,BC †

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Introduction The continuous production of bubbles from submerged capillaries is common in many technical applications such as water treatment, 1 metallurgy, 2 and in blood oxygenators. 3 The production of bubbles from capillaries is also useful for laboratory research and is central to techniques such as the maximum bubble pressure tensiometry. 4 Aside from the continuous production of bubbles from a capillary, it is also useful to study individual bubbles supported on the tip of a capillary with precise control of the bubble volume. Individual bubbles are used in measurements of interfacial dilatational rheology, 5–8 as well as for thin-film drainage experiments. 9–11 The practical applications, together with the rich mechanics of bubbles have attracted researchers since the time of Laplace. 12 For example, researchers have investigated the shape of static and rising bubbles, 13–15 the size and rate of bubble production, 16 coalescence phenomena, 17,18 foam formation, 19 and the rheological properties of air/liquid interfaces. 7,20 As bubbles are composed of a gas phase (by definition), it is important to consider the impact of compressibility. In some cases, the effects of compressibility are not dominant and can be safely neglected, such as predicting the bubble volume at which buoyancy will lead to separation from the capillary, 16 while in other cases compressibility can be a key factor such as in the dynamics of growth of small bubbles on capillaries. 13,21 When attempting to perform careful experiments with individual bubbles, the compressibility can make precise control of the bubble volume extremely difficult. This is particularly problematic for rheological studies which are extremely sensitive to changes in the surface area of the bubble. 7,22 One way that the compressibility of the system becomes problematic is that it can cause an instability to occur when the size of the bubble reaches a critical value, inducing a large and nearly instantaneous change in the volume. 21,23 This instability is directly the result of using a capillary to produce or manipulate bubbles and therefore it is important to understand it when employing any of the experimental techniques outlined above. As will be seen in the analysis section below, compressibility results in a hysteresis in an expansion and contraction cycle and is effected by two dimensionless parameters: the dimensionless volume of the system, A, and the dimensionless maximum capillary pressure, Pe. The purpose of this manuscript 2 ACS Paragon Plus Environment

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is to provide a quantitative description of this instability and hysteresis, and to enable researchers to anticipate and/or avoid this instability. Other researchers have been aware of this instability and have attempted to work around it, but to our knowledge, this will be the first comprehensive and quantitative description. One method employed by past researchers to minimize the effect of the compressibility is to reduce the total volume of gas in the system by back-filling the capillary with liquid. 24,25 Other researchers have tried to side-step the issue by controlling the pressure of the bubble (rather than the volume) and keeping the bubble size reasonably far from the point of instability. 26 In this case, controlling the pressure does not eliminate the instability, but it does provide a more convenient method for experimentally controlling the radius of curvature of the bubble. The analysis in the present work is on the case of a bubble protruding from a capillary and undergoing changes in volume with the edge of the bubble pinned at the inner diameter of the capillary as shown in Fig.1a. The changes in volume are assumed to occur quasi-statically such that the effects of inertia and viscous resistances are negligible. The analysis reveals that, depending on the system parameters such as the total volume of gas and the capillary diameter, there are two regimes of volume change. In the first regime, the bubble will experience a rapid change in volume - associated with an instability - upon expansion and contraction. Even though instabilities have been observed by researchers before, it has not been realized that the critical bubble radius at which the instability occurs is not the same during expansion and contraction. In other words, a hysteresis is present. In the second regime, the bubble will continuously expand and contract through all bubble sizes accessible prior to separation. Beyond this size, buoyancy overcomes the surface pinning and causes the bubble to detach from the capillary. In this second case, even though the instability is avoided, the analysis reveals that there is still not a one-to-one correspondence between the change in the volume of the bubble and the rest of the system (the capillary and other plumbing fixtures). This result has important implications when attempting to dynamically vary the bubble size with a syringe pump and will be discussed in detail later. In both regimes, experimental results are found to agree quantitatively with the predictions of the analysis.


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Methods Experiments The experimental setup is shown in Fig.1b and consists of a gas-tight syringe (Hamilton, Reno, NV) that is connected via small, inner-diameter tubing (1/16" OD, 0.03" ID, IDEX Health & Science, Rohnert Park, CA ) to a pressure transducer (Omega PX409, OMEGA Engineering Inc, Stamford, CT) and a blunt-tipped syringe needle (Vita Needle Company, Needham, MA). The syringe is actuated using a syringe pump (Harvard Apparatus Pico Plus 11, Harvard Apparatus, Holliston, MA) and the needle is submerged in a custom-built chamber with transparent side walls and a volume of about 6 ml. The chamber is illuminated on one side using a lamp with a fiber optic cable, while a CCD camera (Thor Labs DCU223C, Thorlabs Inc, Newton, NJ) is positioned on the opposite side for monitoring the size of the R bubble. MATLAB is used to control the syringe pump while simultaneously measuring the bubble size

and pressure. Experiments are performed by first creating a bubble at the tip of the needle that forms a spherical cap much smaller than a hemisphere. The syringe pump is then operated at a fixed flow rate until the bubble is much larger than a hemisphere, but small enough to avoid separation due to buoyancy. After increasing the size of the bubble, it is held at this volume for 10 seconds and subsequently the volume is reduced at the same rate until the original bubble size is reached. The bubble volume is monitored using the CCD camera at 30 Hz and with a spatial resolution of 4.8 µm/pixel. The pressure inside the bubble was sampled at 40 Hz with a repeatability of ± 0.85 P a. The experiments were performed using two different configurations, to access the two different regimes of expansion and contraction. In one, a 500 µL syringe was coupled with a 16 gauge needle (OD: 1.651±0.013 mm, ID: 1.194±0.038 mm) and the syringe pump was operated at a flow rate of 500 nL/s. In the other 50 µL syringe was coupled with a 19 gauge needle (OD: 1.067±0.013 mm, ID: 0.686±0.038 mm) with the syringe pump operated at 100 nL/s. All of the experiments were conducted at atmospheric pressure (101325 Pa) using deionized water (with a surface tension of ∼72 mN/m) as the liquid in the chamber.


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Figure 1: (a)Schematic of a bubble pinned on a capillary showing the nomenclature used for modeling. Vb is the bubble volume and Vd is the dead volume that fills the remainder of the system including the syringe pump, pressure transducer, and tubing. (b) Schematic representation of the experimental setup. (c) An image of the bubble obtained using the CCD camera during an experiment.

Modeling In this section we develop a quasi-static model for a compressible, spherical bubble evolving on a capillary where it is pinned at the inner edge of the capillary opening. The bubble is contiguous with an additional volume of air within the capillary Vd (Fig.1.c) that can be adjusted (as is common in experiments with syringe pumps). As a result, this model addresses the intermediate regime of bubble formation defined by Kumar et al. 16 in which both the rate of volume increase of the bubble and the pressure inside the capillary are changing. The model is expected to remain valid as long as the Bond number (Bo = ∆ρga2 /σ), the capillary number (Ca = µl q0 /σa2 ), and the Weber number (We = ρl q02 /σa3 ) are 1 ; where ∆ρ is the density difference between the liquid density ρl and the gas density ρg , g the acceleration due to gravity, a the radius of the capillary, σ the surface tension coefficient, q0 the mean flow rate and µl the liquid viscosity. The parameters influencing the volume of a pinned bubble on a capillary are: 27


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Vb = f (q, a, ρl , µl , σ, ρg , µg , K, Vd , g, Hs ),

(1)

where q is the flow rate through the capillary orifice, a the radius of the capillary, K the orifice constant, Vd the dead volume filling the remainder of the system (syringe pump, pressure transducer and tubing), Hs the hydrostatic head and µg the viscosity of the gas. For a quasi-statically evolving bubble, the total volume (V ) of gas in the system, which is the sum of Vb and Vd , can be related to the pressure (P ) in the system through the isothermal relation P V = C, implying that the pressure can also be expressed in terms of the variables in Eq.1. The isothermal relation holds provided that the mass transfer across the interface and the pressure drop throughout the dead volume of the system are both negligible. When the pressure drop is small, we can also neglect the influence of K and µg on the bubble volume. 14,28 The pressure inside an evolving bubble is related to the variables in Eq.1, using the well-known Rayleigh Plesset equation. The pressure inside large bubbles is also affected by the variation of the hydrostatic pressure along the height of the bubble. Taking into account this hydrostatic pressure and non-dimensionalizing the resulting pressure by the maximum capillary pressure

1 1 dR (P − P0 )a = + Bo H + Ca 2 2σ R R dτ " 2 # R d2 R 3 dR + We + , 2 dτ 2 4 dτ

2σ a

yields:

(2)

where P0 is the static pressure at the apex of the bubble, R and H are the radius of curvature and the height of the bubble non-dimensionalized by the radius of the capillary a, and τ the time non-dimensionalized by

a3 . q0

As mentioned above, for this analysis we assume Ca , We , and Bo are 1. In this limit, using Eq.2

in the isothermal relation, and applying the formulas for a spherical cap to obtain the bubble radius and volume in terms of the bubble height we get: P0 + 4σ

h i h π 2 2 V + h(3a + h ) = C, d h2 + a2 6


(3)

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where h is the dimensional bubble height, σ is the surface tension, Vd is the dead volume in the system, and C is a constant. In an experiment, the dead volume changes with time. We define the dead volume at the point when the bubble volume is zero to be Vd,0 . The static pressure when Vd = Vd,0 is P0 , and this defines C = P0 Vd,0 , which remains constant throughout an experiment. Applying this definition of C and non-dimensionalizing yields:

h i H 1 + Pe 2 Ved + A(3H + H 3 ) = 1, H +1

(4)

where

H=

Vd h e πa3 e 4σ ; Vd = . ; A= ; P = a Vd,0 6Vd,0 P0 a

This equation provides a relationship for changes in the height of the bubble in response to changes in the dead volume within the system for a given value of A and Pe. Note that because Eq.4 is implicit in H, it is possible for multiple bubble heights to satisfy this relationship. Also, when the effect of surface tension is negligible (Pe 1), Eq.4 reduces to the incompressible limit. In most experiments, the dead volume changes linearly in time according to Vd = qt, or in general as R Vd = qdt when flow rate is not a constant. This dependence can be accounted for by simply differentiating Eq.4 with respect to time to give:

dH Q (P H 2 + P + H) A (H 2 + 1) = dt (P H 2 + P + H) A (H 2 + 1)2 − V` (H 2 − 1)

(5)

where,

V` = 1 − A Q=

Z

t

Qdt + A 0

H (3 + H 2 ) 3

1 dVed . A dt

(6) (7)


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}

(a)

}

Super cr i t i calCase I nf usi on Suct i on

Subcr i t i calCase I nf usi on Suct i on

} }

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(b)

Figure 2: (a) Pressure traces superimposed with bubble shapes from an experiment in the subcritical regime using a 16 gauge needle, syringe volume of 500µL, and a constant flow rate of 0.5µLs−1 , showing identical expansion and contraction. (b) Pressure traces superimposed with bubble shapes from an experiment in the supercritical regime using a 19 gauge needle, syringe volume of 100µL, and a constant flow rate of 0.2µLs−1 , showing sharp changes in pressure and asymmetrical expansion and contraction. Eq.5 is left dimensional in time as there is no natural scale for time. Furthermore, Eq.5 can be used to predict the effect of compressibility on dynamic bubble experiments as long as the pressure is dominated by surface tension in Eq.1, i.e in the limit of low Ca and We .

Numerical Solutions There is no general solution in terms of radicals for Eq.4, which is a fifth order polynomial in H. This can be shown using results from the Galois group theory. 29 Hence the roots of Eq.4 are determined using an R inbuilt MATLAB function that determines the roots of the polynomial by computing the eigenvalues of

its companion matrix. The solutions of Eq.5 are obtained using the 4th order Runge Kutta routines in Matlab.


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Super c r i t i c al Regi me

Subc r i t i c al Regi me

Ac c es s i bl eSt at es

I nac c es s i bl eSt at es

( b)

( a) I nf us i onandWi t hdr awal

( c ) I nf us i on

Wi t hdr awal

Figure 3: The accessible states of a bubble in the two regimes of bubble formation. (a) In the subcritical regime all bubble shapes are accessible. (b) In the supercritical case during infusion, the bubble undergoes a jump (∆Hinf usion ) between the stable states. The inaccessible intermediate states are shown by dashed lines. (c) In the supercritical case during withdrawal, there is another jump (∆Hwithdrawal ). However, the magnitude of this jump is always less than that during expansion, giving rise to the bubble shape hysteresis.

Results and Discussion Experiments Experiments were conducted with bubbles expanding and contracting on capillaries, and the volume and pressure within the bubble were recorded as functions of time. For the purposes of discussion, we present results from an experiment conducted using a 16 gauge capillary, which will hereafter be referred to as the subcritical case, and another conducted using a 19 gauge capillary, which will be referred to as the supercritical case. The pressure plots of the experiments shown in Fig.2 clearly illustrate the two distinct regimes for bubble expansion and contraction. In the subcritical case, the bubble expands and contracts reversibly: there are no sharp changes in pressure and bubble volumes, and during contraction, the pressure and volume retrace the expansion profiles exactly. However in the supercritical case, there are sharp changes (or “jumps”) in the pressure and volume, and during contraction the pressure profiles and bubble volumes


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deviate from the expansion profiles. Thus, in the supercritical regime the bubble exhibits a hysteresis, with the jump in pressure during expansion being much larger than the jump during contraction. The nature of the transition from subcritical to supercritical behavior is shown in Fig.3. In the subcritical regime, bubbles can be expanded and contracted in volume continuously and there is a unique bubble volume corresponding to a given dead volume. In the supercritical regime, the relationship between bubble volume and dead volume is complicated, with certain values of bubble volume becoming unstable depending on whether the bubble is infused or withdrawn. This is indicated in Fig.3(b) and (c), by the dashed bubble curves to represent unstable states. Upon infusion, bubbles in this regime will immediately pass through the unstable states as they transition from one stable state to another. During withdrawal, a similar jump between stable states will also occur, but, as shown Fig. 3(c) the jump during withdrawal is smaller in magnitude. The magnitude of the jumps between stable states can be characterized by either measurements in transitions in bubble height or pressure. In Fig. 3(b) and (c), these height transitions, ∆Hinf usion and ∆Hwithdrawal , respectively, are shown. The magnitude of ∆Hinf usion is always greater than ∆Hwithdrawal giving rise to the bubble shape hysteresis. Unlike a rubber balloon, where hysteresis exists due to the large deformation of the rubber membranes and consequent change in the elastic properties, 30,31 the origin of bubble shape hysteresis in the supercritical regime is a result of the compressibility of the gas.

Comparison to Model Calculations The existence of these two regimes in the bubble formation is predicted by Eq.4. This can be explicitly shown by comparing the results from Eq.4 with that from the experiments, and for that we need to determine the value of the dead volume in our experimental system. However, we cannot easily measure the dead volume. Instead, we can take advantage of Eq.4 to compute the dead volume from measurements of the bubble volume. We know that changing Ved results in a change in the bubble volume, according to the following algebraic rearrangement of Eq.4: Ved (H) =

H2 + 1 − A 3H + H 3 H 2 + P˜ H + 1 10

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(8)

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Hence we can develop a formula for Vd,0 based on a known change in dead volume required to change the volume of a bubble from one height H1 to another height H2 : ∆Ved = Ved (H1 ) − Ved (H2 ). After some algebra we arrive at the following:

Vd,0 =

∆Vd +

πa3 [3 (H1 − H2 ) + H13 6 H12 +1 H22 +1 − 2 2 ˜ H1 +P H1 +1 H2 +P˜ H2 +1

− H23 ]

(9)

Using this formula, we calculated the initial dead volumes for experiments in the subcritical and supercritical cases to be 1.2832 ml and 0.5297 ml respectively. With the initial dead volumes known, Eq.4 is used to plot the predicted bubble volume as a function of the change in dead volume due to actuating the syringe. The results of these calculations are compared against experimental data in Fig.4. For the subcritical regime, there is excellent agreement between the predicted values and the measured values for all dead volumes considered. For the supercritical regime also there is excellent agreement, but the situation is more complicated because of the hysteresis phenomenon described in Fig.3. In this regime, multiple values of bubble sizes are predicted for a given dead volume. In the supercritical case, when the bubble is small there is only one predicted value for the bubble volume. Then, as volume is infused into the bubble from the syringe, the bubble volume follows the lowest of the three predicted values. Eventually, the region with three predicted values ends (when the radius the bubble is approximately equal to the radius of the capillary) and a further increase in volume results in the rapid jump to the only volume predicted for that dead volume. Following the jump, the volume can be withdrawn from the bubble by reversing the motion of the syringe, but the bubble volume now follows the largest of the predicted values. Again, the bubble shrinks until it reaches the end of the region where three values are predicted and then jumps to the only predicted value below that dead volume. Theoretically, all three predicted values represent equilibrium states of the bubble for a given dead volume, but the intermediate values are unstable states. The stability of a given equilibrium state can be evaluated by considering the response to a small perturbation in the bubble volume. An equilibrium state is stable if the perturbation results in a net change in pressure which brings the bubble back to its unperturbed state, i.e a perturbation increasing


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Figure 4: Comparison of solutions to Eq.4 against experiments. Two cases are shown, a subcritical case (A = Aref = 8.6 × 10−5 , Pe = 4.8 × 10−3 ) and a supercritical case (A = 3.98 × 10−5 , Pe = 8.2 × 10−3 ). The subcritical solution is shown by and the supercritical solution is shown by . The subcritical experimental data are shown as circles and the supercritical data are squares. The open symbols are for 3Vb vs 2πa infusion and solid symbols are for withdrawal of gas. Stability curves ( dP 3 ) corresponding to the dh > 0. upper and left axes respectively - shown as dash-dot lines - confirm that the unstable states have dP dh A ref ∆V˜d is scaled by A for ease of visualization of the different experiments on the same plot. the bubble volume should be accompanied by a net decrease in pressure and vice versa. The net change in the pressure due to a perturbation of the bubble volume (equivalently a perturbation on the bubble height at a given Ved ),

dP , dh

is the sum of the change in Laplace pressure exerted by the bubble and change

in pressure due to change in volume of the system, and is negative for stable equilibrium states. The stability curves are also shown in Fig.4 as the dash-dot curves. In the subcritical regime of this figure, all the equilibrium states are stable ( dP < 0). However in the supercritical regime, dh

dP dh

turns positive

in the region where multiple steady states are predicted. Experimentally, we see that the bubble in the supercritical regime is observed to jump exactly when

dP dh

= 0. The subsequent stable equilibrium state

to which the bubble jumps is such that the magnitude of the jump during expansion is much larger than that during contraction, resulting in the hysteresis. The bubble shape hysteresis, for the purposes of this paper, is quantified as the difference in the 12 ACS Paragon Plus Environment

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2 2

4

0.9 16

1.5

0.8 32

1

8

0.7 2

4

0.5 0.6

16

0

0.5

−0.5 32

log(P˜ )

0.4

−1

8

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4

−1.5

2

0.3

16

0.2

−2 −2.5

8

0.1 4

−3 −6

2

−5

−4

−3 log(A)

−2

−1

0

0

Figure 5: The color map shows the magnitude of the bubble shape hysteresis in terms of the bubble pressure, defined as the difference of the jump in pressure during the bubble expansion to that during bubble contraction . The bold line demarcates the two regimes of bubble formation, and is obtained from the solution of Eq.10. The contour lines and the corresponding values indicate the magnitude of the jump experienced by the bubble during expansion (∆Hinf uson ) and may be used to determine whether a bubble remains attached to the capillary after the jump. magnitude of the jump in the bubble pressure or volume during expansion to that during contraction. The bubble pressure ( H4σH 2 +1 ) can be calculated before and after a jump, and the difference gives the magnitude of the jump in bubble pressure during the transition. The difference in the magnitude of the jump in bubble pressure across infusion and contraction gives the magnitude of hysteresis in terms of the bubble pressure. Similarly the magnitude of hysteresis can be obtained in terms of the bubble volume. Here we choose to show the hysteresis in terms of the bubble pressure as a function of A and Pe in Fig.5, as it supports the pressure plots shown previously (Fig.2). Consistent with our experiments, we see that hysteresis exists only in the supercritical regime of bubble formation. Further, the magnitude of the hysteresis increases monotonically when A is decreased at a given Pe. While as Pe is varied at a given A, the magnitude of the hysteresis is seen to go through a maximum. This is due to the fact that at large Pe, surface tension dominates restricting the magnitude of the jump, and at very small Pe, the static pressure dominates reducing the compressibility of the gas (bulk modulus increases).


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The sensitivity of the system to these parameters can also be viewed by plotting the solutions to Eq.4 for several values of A and Pe (Fig.6). For the purposes of illustration, we have chosen an initial bubble volume similar to those in our experiments. Qualitatively we see that for values of A (for a given Pe) larger than a critical value, there always exists a unique equilibrium state for a given syringe volume. This corresponds to the experimental observations in the supercritical case for which the value of A (= 8.6 × 10−5 ) was larger than the critical value (5.8 × 10−5 ) at that Pe =

1 . 211

Further, in the subcritical regime when the surface tension becomes negligible (P → 0)

and/or the dead volume becomes comparable to the bubble volume (A → O (1)), the bubble expansion approaches the “incompressible limit” where the gas inside the bubble and dead volume behaves as an incompressible fluid. In that case, any change in dead volume will result in an equal change in the bubble volume. On the other hand for values of A (for a given Pe) less than a critical value, there are multiple equilibrium states for some values of the syringe volume. This corresponds to the experimental observations in the subcritical case, where A (= 3.98 × 10−5 ) was less than the critical value (1 × 10−4 ) at that Pe =

1 . 120.7

Hence the regime of bubble formation can be determined simply by comparing the value

of A to its critical value at that Pe. Estimate for the critical value of A The value of A which demarcates the two regimes of bubble formation, which will here on be referred as Acrit , can aid in designing experiments to obtain bubble expansion in the desired regime. To obtain the value of Acrit , we can solve for the value of A (at a given Pe) at which Eq.4 has a saddle point (Ved wrt H). Physically this saddle point corresponds to an equilibrium configuration of the bubble that is neutrally stable to perturbations. Solving for this saddle point we obtain,

Acrit =

Pe(h2s − 1)

3(h2s + 1)(h2s + 1 + Pehs )2

where hs satisfies, 2h5s + Peh4s − 2h3s − 2Peh2s − 4hs − Pe = 0


(10)

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10 9 8 7 6 3Vb 2πa3 5

4 3 2

A = 1.0 × 10−3 , P˜ =

1 141

A = 8.6 × 10−5 , P˜ =

1 141

A = 7.0 × 10−5 , P˜ =

1 141

A = 8.6 × 10−5 , P˜ =

1 80

−5

A = 8.6 × 10

1

, P˜ =

1 300

Incompressible 0 0

0.5

1

1.5

(∆V˜d )

2

Aref A

2.5 −3

x 10

Figure 6: The locus of the equilibrium states of the bubble for different values of A and Pe. The solid lines show the consequence of changing A at a fixed Pe, while the hashed lines have fixed A and varying 1 Pe. With reference to the critical case of A = 8.6 × 10−5 , Pe = 141 , we see that an increase in A at a fixed e P results in a subcritical bubble expansion, while a decrease of A results in a supercritical expansion with hysteresis. Similar behavior is observed when varying Pe. In the limit of A → O (1) and/or Pe → 0, the bubble expansion approaches the incompressible limit. We see that Acrit is a function of hs , which is the only positive root of the above equation at a given Pe. In the limit of small and large Pe this gives: Pe Pe 1 ; 81 0.04044 ≈ ; Pe 1 Pe

Acrit ≈

(11)

Acrit

(12)

Eq.11 can be rearranged to obtain the Bubble Stability Number (BSN) defined by Liggieri et.al. 23 For experiments conducted at atmospheric pressure and using standard capillaries, Pe is usually very small and hence Eq.11 gives a very good estimate for Acrit . Based on the above calculations for Acrit , an air bubble in clean water with a capillary of diameter of 1 mm, must have a dead volume (Vd ) less than 900 µL to avoid the instability. In dimensional terms,


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Eq.11 predicts that the critical dead volume scales as a4 . Hence to have a stable bubble on a capillary of half the size the dead volume must be less than

1 th 16

of the previous case i.e 56 µL. Therefore, it

becomes increasingly difficult to avoid the supercritical regime when studying small spherical bubbles. The problem is exacerbated by the addition of a pressure transducer to the system, which significantly increases the dead volume.

Estimate for the jump In the second regime of bubble formation, the magnitude of the jump during bubble expansion determines whether the bubble remains pinned on the capillary or separates from the capillary due to buoyancy. Since many experiments require bubbles to remain pinned on the capillaries, we calculate the magnitude of the jump from Eq.4. The magnitude of the jump is obtained by solving for the local extrema of Eq. 4. The difference between the height of the bubble at the extrema to the other predicted height of the bubble at the same volume is then the magnitude of the jump. Iso contours of the jump during infusion, ∆Hinf uson , is shown in Fig.5. Similar to hysteresis, the magnitude of the jump increases monotonically as A decreases for a fixed value of Pe. Hence at very small values of A, usually corresponding to large a dead volume, compressibility causes the bubble to undergo large jumps in volume, and will separate from the capillary due to buoyancy during expansion.

Dilational Rheology Measurements There are many experiments which require bubble volumes to be oscillated in time. An important class of experiments are rheology measurements where the dilatational mechanics of air/liquid interfaces are examined. 5 In these experiments, it is important to faithfully follow the phase relationship between oscillatory changes in interfacial area to interfacial stresses. It is evident from the analysis and experiments described above that compressibility can introduce an unintended complication in this relationship. This will be explored here for the simple case of air/liquid interface that is characterized by constant surface tension. 16 ACS Paragon Plus Environment

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8 A = 1 × 10−2 A = 1 × 10−3 A = 1 × 10−4

7

A = 6.66 × 10−5 A = 5.83 × 10−5 Incompressible

6

5 1 dVb q dt

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4

3

2

1

0 0

1

2

3

4

5

6

7

8

9

10

3Vb 2πa3

Figure 7: The ratio of temporal variation of the bubble volume to the constant pump flow rate as a function of the non dimensional volume of the bubble for different A below the critical value (Acrit = 5.8 × 10−5 ) 1 . The effects of compressibility are dominant when the bubble is a spherical cap and when at Pe = 211 the bubble expands past a hemispherical shape. At the larger volumes, the bubble can be treated to be incompressible. Consider the response of a bubble as its volume is increased at a rate

dVb . dt

This rate, when normalized

by the infusion rate q, is shown plotted in Fig.7 as a function of the normalized bubble volume. In this plot, Pe is held constant at a value of

1 , 211

while A is varied. In most experiments it is desirable to operate

under conditions where the normalized rate of volume change is relatively constant and close to unity. This plot indicates that compressibility of the gas can lead to strong deviations away from unity when A becomes sufficiently small and when the bubble smaller than a hemisphere, even when A is large. Compressibility can also cause problems with using a syringe pump to impose other types of dynamic strains to a bubble interface. For example, experiments involving sinusoidal area strains are very common. 6,32 As previously mentioned, compressibility can lead to a deviation in the actual volume growth of a bubble from the imposed flow rate. However, it is possible to impose a flow rate which can compensate for this deviation. This flow rate is obtained by solving a differential form of Eq.4 with the desired constraints and is shown in Fig.8 (for cases with low Ca and W e when the quasi-static assumption holds). For


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20

for Sinusoidal Oscillations

Constant Area Strain (C) Constant Area Strain (I) Sinusoidal Area Strain(C) Sinusoidal Area Strain(I)

0

2q πa3

for Area Strain, Volume Growth rate

40

2q πa3

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0 0

1

2

3

4

5

6

7

−1 8

Time (s)

Figure 8: The theoretical flow rates to compensate for compressibility when producing a sinusoidal area 1 strain and a constantly increasing area strain for Pe = 211 , A = 8.6 × 10−5 (similar to those in our experiments). The initial bubble height is chosen as H = 0.4 for the constant area strain (0.5s−1 ) 1 Hz. Note: In the calculations. H = 0.8 for the sinusoidal oscillations of amplitude 0.2 and frequency 2π legend C denotes calculations accounting for compressibility and I denotes the incompressible case for comparison. sinusoidal oscillations, the imposed flow rate which compensates for compressibility is larger in magnitude and changes more rapidly during the expansion phase than the compression phase, as compared to that obtained assuming incompressibility. Constant area strain rate experiments can also be performed using bubbles. In the incompressible limit, constant area strain rate can be obtained by an exponentially increasing flow rate. However when compressibility is accounted for, the required flow rate is a complex function of time (Fig.8), which as the bubble size becomes large, asymptotes to the incompressible limit. Practically speaking, these flow rates could be implemented using commercial syringe pumps, except at low values of A when the required flow rates can become very large. Aside from complicating the precise control over the interfacial strain, compressibility influences the size of bubbles that can be studied. When the bubble undergoes a jump, not all sizes of bubbles can be utilized. The size of bubbles that cannot be formed fall in the region where 18 ACS Paragon Plus Environment

dP dh

≥ 0. On the other hand,

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the jump can be taken advantage of to perform large step strain experiments where the change in volume is very close to a theoretical step strain. Moreover, the magnitude of the jump and hence that of the step strain, can be easily calculated once A and Pe are determined as done in Fig.5. It is important to note that the above analyses were performed for systems with constant surface tension in the limit of small Bond, capillary, and Weber numbers. However, bubble pressure tensiometers are often used to study the dynamics of complex interfaces under conditions where the capillary and Weber numbers may not be negligible. In addition, the surface stresses may be viscoelastic and depend on the strain rate, in which case the functional form of the pressure (as in Eq.1) will have additional terms to account for this. Future work needs to be carried out to account for these effects.

Conclusion We have experimentally and theoretically analyzed the effects of compressibility during the expansion and contraction of bubbles on capillaries. We have shown through our results that there are two regimes of bubble formation depending on the values of two dimensionless parameters A and Pe. For values of A below a critical value, there always exists a bubble shape hysteresis. The bubble shape hysteresis, unlike other well known hysteresis phenomenon, arises from a relatively simple system - a bubble pinned on a capillary. Like the contact angle hysteresis, the bubble shape hysteresis is another hysteretic phenomenon in which interfaces play an important role. When bubble shape hysteresis exists, not all sizes of bubbles can be formed on the capillary. In order to obtain stable bubbles of all sizes below the separation size, experiments should be conducted with A > Acrit. We have also provided guidance on how a desired strain can be imposed on a bubble for dilational rheology experiments, accounting for compressibility.

Acknowledgement This work was funded by the National Science Foundation under grant number CBET-1435683.


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References (1) Miichi, T.; Hayashi, N.; Ihara, S.; Satoh, S.; Yamabe, C. Generation of radicals using discharge inside bubbles in water for water treatment. Ozone: Sci. Eng. 2002, 24, 471–477. (2) Sano, M.; Mori, K. Bubble formation from single nozzles in liquid metals. Trans. Jpn. Inst. Met. 1976, 17, 344–352. (3) Sutherland, K. M.; Pearson, D. T.; Gordon, L. S. Independent control of blood gas PO2 and PCO2 in a bubble oxygenator. Clin. Phys. Physiol. Meas. 1988, 9, 97–105. (4) Mysels, K. J. The maximum bubble pressure method of measuring surface tension, revisited. Colloids Surf. 1990, 43, 241–262. (5) Kao, R.; Edwards, D.; Wasan, D.; Chen, E. Measurement of interfacial dilatational viscosity at high rates of interface expansion using the maximum bubble pressure method. I. Gas–liquid surface. J. Colloid Interface Sci. 1992, 148, 247–256. (6) Wantke, K.-D.; Fruhner, H. Determination of surface dilational viscosity using the oscillating bubble method. J. Colloid Interface Sci. 2001, 237, 185–199. (7) Lin, G. L.; Pathak, J. A.; Kim, D. H.; Carlson, M.; Riguero, V.; Kim, Y. J.; Buff, J. S.; Fuller, G. G. Interfacial dilatational deformation accelerates particle formation in monoclonal antibody solutions. Soft matter 2016, 12, 3293–3302. (8) Alvarez, N. J.; Anna, S. L.; Saigal, T.; Tilton, R. D.; Walker, L. M. Interfacial dynamics and rheology of polymer-grafted nanoparticles at air–water and xylene–water interfaces. Langmuir 2012, 28, 8052– 8063. (9) Chan, D. Y.; Klaseboer, E.; Manica, R. Film drainage and coalescence between deformable drops and bubbles. Soft Matter 2011, 7, 2235–2264. (10) Hermans, E.; Bhamla, M. S.; Kao, P.; Fuller, G. G.; Vermant, J. Lung surfactants and different contributions to thin film stability. Soft matter 2015, 11, 8048–8057. 20 ACS Paragon Plus Environment

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(11) Frostad, J. M.; Tammaro, D.; Santollani, L.; de Araujo, S. B.; Fuller, G. G. Dynamic fluid-film interferometry as a predictor of bulk foam properties. Soft Matter 2016, (12) Rowlinson, J. S.; Widom, B. Molecular theory of capillarity; Dover Publications, Inc., 2013. (13) Oguz, H. N.; Prosperetti, A. Dynamics of bubble growth and detachment from a needle. J. Fluid Mech. 1993, 257, 111–145. (14) Gerlach, D.; Biswas, G.; Durst, F.; Kolobaric, V. Quasi-static bubble formation on submerged orifices. Int. J. Heat Mass Transfer 2005, 48, 425–438. (15) Longuet-Higgins, M.; Kerman, B. R.; Lunde, K. The release of air bubbles from an underwater nozzle. J. Fluid Mech. 1991, 230, 365–390. (16) Kumar, R.; Kuloor, N. The formation of bubbles and drops. Adv. Chem. Eng. 1970, 8, 255–368. (17) Ata, S. Coalescence of bubbles covered by particles. Langmuir 2008, 24, 6085–6091. (18) Tse, K.; Martin, T.; Mcfarlane, C. M.; Nienow, A. W. Visualisation of bubble coalescence in a coalescence cell, a stirred tank and a bubble column. Chem. Eng. Sci. 1998, 53, 4031–4036. (19) Weaire, D. L.; Hutzler, S. The physics of foams; Oxford University Press, 2001. (20) Wang, Z.; Narsimhan, G. Interfacial dilatational elasticity and viscosity of β-lactoglobulin at airwater interface using pulsating bubble tensiometry. Langmuir 2005, 21, 4482–4489. (21) Hallowell, C. P.; Hirt, D. E. Unusual characteristics of the maximum bubble pressure method using a Teflon capillary. J. Colloid Interface Sci. 1994, 168, 281–288. (22) Liggieri, L.; Attolini, V.; Ferrari, M.; Ravera, F. Measurement of the surface dilational viscoelasticity of adsorbed layers with a capillary pressure tensiometer. J. Colloid Interface Sci. 2002, 255, 225–235. (23) Liggieri, L.; Ravera, F.; Passerone, A. Drop formation instabilities induced by entrapped gas bubbles. J. Colloid Interface Sci. 1990, 140, 436–443.


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(24) Wong, H.; Rumschitzki, D.; Maldarelli, C. Theory and experiment on the low-Reynolds-number expansion and contraction of a bubble pinned at a submerged tube tip. J. Fluid Mech. 1998, 356, 93–124. (25) Frostad, J. M. Fundamental Investigations of Phase Separation in Multiphase Fluids; University of California, Santa Barbara, 2013. (26) Alvarez, N. J.; Walker, L. M.; Anna, S. L. A microtensiometer to probe the effect of radius of curvature on surfactant transport to a spherical interface. Langmuir 2010, 26, 13310–13319. (27) Cliff, R.; Grace, J.; Weber, M. Bubbles, Drops and Particles; Academic Press Inc., New York, 1978. (28) Kovalchuk, V.; Dukhin, S. Dynamic effects in maximum bubble pressure experiments. Colloids Surf., A 2001, 192, 131–155. (29) Lang, S. Algebra; Springer Science and Media, 2002. (30) Ješková, Z.; Featonby, D.; Feková, V. Balloons revisited. Phys. Educ. 2012, 47, 392–398. (31) Merritt, D.; Weinhaus, F. The pressure curve for a rubber balloon. Am. J. Phys. 1978, 46, 976–977. (32) Fruhner, H.; Wantke, K.-D. A new oscillating bubble technique for measuring surface dilational properties. Colloids Surf., A 1996, 114, 53–59.


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Graphical TOC Entry Supercritical Case

380 340 300 260

0

10

20

30

Time (s)

40

50

Infusion

Withdrawal



500



Withdrawal



Infusion



420

Bubble Pressure (Pa)

Subcritical Case

Bubble Pressure (Pa)

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450 400 350 300 250 0

5

10 15 20

25 30 35

Time (s)


Impact of Compressibility on the Control of Bubble-Pressure

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