The formation and growth of a bubble in a liquid is a subject that has been widely investigated. In the present study, bubble growth in carbonated liquids was ...
237
C&ids and Surfaces A: Physicochemical and Engineering Aspects. 85 (1994) 231-253 0 1994 ~ Elsevier Science B.V. All rights reserved. 0927-7757/94/$07.00
Bubble growth in carbonated Chris G.J. Bisperink”, Wageningen Biotechnion,
(Received
Agricultural De Dreyen,
10 December
liquids
Albert Prim University, Department of Food Science, Dairying and Food P.O. Box 8129, 6700 EV Wugeningen, The Netherlunds
1993; accepted
24 December
Physics
Group,
1993)
Abstract The formation and growth of a bubble in a liquid is a subject that has been widely investigated. In the present study, bubble growth in carbonated liquids was investigated. A model cavity was developed at which bubble growth occurred as soon as the liquid became sufficiently oversaturated with gas. A theory based on Fick’s first law was developed to describe this bubble growth at the cavity and to predict the moment of detachment from the latter. This theory was compared with two other available theories by F.C. Frank (Proc. R. Sot. London, 201A (1950) 586), and F. van Voorst Vader, F. Erkens and M. van den Tempel (Trans. Faraday Sot.. 60 (1964) 1170), describing the transport of molecules to a liquid surface driven by a concentration gradient. It appeared that by considering the geometrical changes that take place at the boundary layer during bubble growth, in combination with the penetration theory and by applying Fick’s first law, it is possible to describe the observed bubble growth at a cavity more successfully than by means of the two theories of Frank and van Voorst Vader. Nevertheless, our theory can only be used if no convection is involved. The moment of detachment depends on the perimeter of attachment, the dynamic surface tension of the liquid and the wetting properties of the cavity. The developed theory was successfully applied by using a computer program to calculate the times needed for very small consecutive increases in bubble radius. In this way, transient phenomena such as bubble growth and detachment could be predicted Key words: Bubble growth;
Carbonated
liquids
Introduction The presence of a foam in a carbonated beverage is, in the first instance, dependent upon bubble formation. The formation and growth of bubbles has been investigated in many studies. Blander [l], Ward et al. [2], Winterton 131, Wilt [4], Ciholas and Wilt [ 51, Cole [ 61, Tao [ 71, Lubetkin and Blackwell [ 81 and Epstein and Plesset [9] have presented studies concerning bubble formation induced by boiling to form vapour bubbles, or by pressure release to form gas bubbles. The latter is investigated in this study, since CO2 bubbles are formed in carbonated beverages, e.g. beer and champagne. The gas phase in these drinks can have a considerable effect on mouthfeel and flavour
*Corresponding
author.
release, and, furthermore, the size distribution and the number of bubbles formed per unit of time determine the appearance and the stability of the foam [lo],
amongst others. According to Ronteltap a wide bubble-size distribution promotes a
high extent of disproportionation, i.e. interbubble gas diffusion as a result of differences in Laplace pressure between bubbles of different sizes leading to foam coarsening. Furthermore, Ronteltap [lo] mentioned that if small bubbles are formed and consequently the fraction of liquid in the foam is high, this contributes to a larger creaminess of the foam. In the case of champagne, a continuously reforming foam ring on the liquid surface, the so-called “colorette” is an important feature of this product. Not only is the number of bubbles formed per unit of time of importance, but also the size distribution
that determines the behaviour, appearance and colour of the colorette. Additionally, the quality of
lowers the free activation
the champagne
is often related
bubbles
in the glass. In this context,
liquid interface A schematic
formed
to the size of the it is
found that a champagne that has received a judgement of being a high quality champagne mostly produces
small
bubbles
on the glass wall, which
then detach from it. At high bubble formation rates, bubble trains can be observed at several sites on the glass wall. It is clear that in the overall appearance of beer and champagne, and also in carbonized beverages in general, bubble formation plays an important role. The initial bubble size distribution in a foam depends on the history of the bubble formation, i.e. the number of bubbles per unit of time, the shape and wetting properties of the cavities, the oversaturation of the liquid with gas, the rheological surface properties of the liquid and the velocity and direction of the flow of the liquid surrounding the bubble. Therefore, it was the aim of this study to investigate the physical mechanism of bubble formation, because it has a direct influence on the appearance of the foam and therefore on its acceptance by the consumers. Theoretical There are, in principle, two mechanisms for bubble formation that can be distinguished: homogeneous and heterogeneous nucleation of bubbles. Theories and experiments performed to investigate these mechanisms in the case of pool boiling are discussed in a review by Blander [ 11. Wilt [4] has predicted that at a supersaturation of about 5, i.e. five times the saturation concentration of CO, at 1 atm, which is typical of a carbonated beverage, bubbles can only be formed by heterogeneous nucleation since homogeneous nucleation requires oversaturation of lo3 or more. In the case of heterogeneous nucleation, the bubbles grow from a catalytic site by which they overcome the energy barrier for bubble formation. This catalytic site can be a gas pocket in a cavity in a solid surface, for example the glass wall or solid particles in the liquid bulk. The existence of such a gaseous phase
tion to negative
energy for bubble
values even because
forma-
the new gas-
is already present. drawing of the process
of bubble
growth out of a gas pocket is presented in Fig. 1. Due to (amongst others) the geometry of this site. the wetting properties and the surface tension of the liquid, the free activation energy for nucleation becomes lower than in the case of homogeneous nucleation. Therefore, due to a lowering of the energy barrier, heterogeneous nucleation can occur at much lower oversaturations of the gas in the liquid compared with homogeneous nucleation. These gas pockets may be present when pouring liquid into a dry glass when a volume of the gas phase is entrapped in a cavity in the glass wall. The stability of these gas pockets depends on the earlier mentioned properties of the cavity and the liquid in terms of geometry, wetting and surface tension. A theory for heterogeneous nucleation was developed by Ward et al. [2,11] and Ward and Levart [ 121 who described the relationship between the pressure at which nucleation occurs and the gas concentration in the liquid. They concluded that the radius of a gas pocket in a wetted conical-shaped cavity must have at least a minimum critical size to allow bubble growth. Gas pockets with a smaller radius than the critical one will dissolve instead of grow. The critical radius will be smaller when the oversaturation of the gas in the liquid is higher. Therefore, in principle. gas pockets may dissolve when the total pressure on the system is increased. If these gas pockets have a radius equal to or larger than the critical radius, they will grow out into a bubble. Furthermore, the stability of a gas pocket in a cavity depends on the wetting properties of the cavity’s wall by the liquid. In the case of a conical-shaped cavity, Blander [ I ] concluded that the contact angle must be greater than half the apical angle of the cavity for the gas pocket to be trapped in the cavity. Therefore, a sharper cavity promotes the stability of a gas pocket. The stability of the gas pocket is promoted when the solid surface is poorly wetted by the liquid. Wilt [4] and Ciholas and Wilt [S]
C.G.J. Bisperink
and A. Prins/Colloids
Fig. 1. A schematic
Surfuces A: Physicochem.
representation
of a bubble
put forward a model for the heterogeneous nucleation of a gas bubble in a conical cavity. They suggested that heterogeneous nucleation of bubbles in depressurized beverages depends on the contact angle between the liquid and the nucleation site, and the shape of this site. According to them, a cavity containing a gas pocket, especially, meets the optimal conditions for bubble formation. Therefore we assume that in beverages such as beer and champagne the bubbles grow out of cavities containing a stable gas pocket. The growth rate of a bubble is determined by several physical parameters. Since bubble growth is a relatively rapid process, it is assumed that both diffusion and convection determine the growth rate of the bubbles. These processes are influenced by the concentration and solubility of the gas, which in turn determine the gas diffusion into the bubble. Furthermore, the temperature and pressure will affect the bubble growth rate because these parameters determine the excess in the concentration of the gas, i.e. oversaturation of the gas in the liquid. In this study, two existing theories that describe gas transport to a surface driven by a concentration gradient between the surface and the liquid bulk
Eng. Aspects 85 (1994) 237-253
growing
239
at a cavity in which the liquid wets the solid
were tested regarding their ability to describe bubble growth. Furthermore, a newly developed theory that describes bubble growth and is based on Fick’s first law was compared to measured growth rates of a bubble. The bubble was situated on a cylindrical model cavity in a liquid oversaturated with gas, as will be discussed later. The comparisons were made by using the experimentally measured fluxes and the fluxes calculated by means of the existing theories. The experimental fluxes were obtained mass equation
by using
dVb Pi+l AiQi = z 7 I
a conservation
of
(1)
where Ai is the bubble surface area available for gas transfer at time i (m’), Bi is the volume transport of gas per unit of bubble surface area per second at time i (m3 mP2 s-l), dVb/dt is the bubble volume growth rate (m3 s-l) and Pi+l/Pi is the ratio representing the change of pressure as a result of the change of bubble radius during growth. This contribution is defined as 2Ydyn pi,
pi+1
=
Ri, Ri+l
+pgh
>
+P
(2)
where Pi and Pi + , are the pressures (Pa) inside the bubble at times i and i + I, respectively, ydy,, is the dynamic surface tension in expansion (N mm-‘) since the bubble
surface is expanded
during growth,
Ri and Ri, 1 are the bubble radii (m) at times i and i + 1, respectively, /I is the density of the liquid due to gravity (kg m -3), g is the acceleration (m se2), h is the height of the liquid column above the bubble (m) and P is the external pressure during bubble growth (Pa). In Eq. (2) the Laplace pressure and the hydrostatic pressure have been added to the external pressure to obtain the total bubble pressure. This correction is necessary in order to determine the correct volume of the bubble that depends on the total bubble pressure, and is essential especially at a very small bubble radius, since the Laplace pressure increases with decreasing bubble radius. The change in volume of a bubble that is attached to a capillary can be calculated using Eq. (3), assuming the bubble has a spherical shape as shown in Fig. 2. d~;_i+,,
=37c[R~+1(2+3~~~/?i+l-co~3/ji+l) - R;( 2 + 3 cos /$ - cos3 Ii)]
(3)
where dVP,,,+I, is the change in bubble volume (m3) in the time interval between i and i + 1, and /3 is defined in Fig. 2. In Eq. (3) a correction has been made for the excluded volume of the spherical segment of the bubble that is present within the cavity, as can be seen in Fig. 2. The concentration difference between the bubble surface and can be calculated using
at time the liquid
t= i bulk
where Aci is the concentration difference of CO, between the liquid bulk and the bubble surface (m3(C0,) rn-“( liquid)), cco, is the concentration of CO2 in the liquid bulk (mol m -“). MV is the molar volume of the CO, in the gaseous phase
Fig. 2. A schematic drawing of a growing bubble at a cavity with the various parameters used in the calculation of the bubble growth rate using the theory developed.
(m3 molt ‘) (MT/ = 0.0245 m3 mol-’
at 293 K and
1 atm), H is the Henry’s law constant (Pa mol-’ kg) (H = 2.9688 x IO6 Pa mol-’ kg for CO, in H,O), and P* is the pressure during saturation with CO, (Pa). Considering the bubble surface during bubble growth, it can be concluded that the former is expanded and is continuously moving towards the liquid bulk, which is the source of the oversaturated gas that causes the bubble to grow. Frank [ 131 described the bubble growth in terms of pure diffusion. taking into account the fact that the bubble surface is moving towards the liquid bulk. This should result in an increase of the gas flux into the bubble due to the diffusion distance becoming smaller. He introduced a dimensionless radius s as presented in Eq. (5), which
C.G.J. Bisperink and A. PrinsiColloids
Surfaces A: Physicochem.
includes the effect of the moving surface on the gas flux. In this equation, Frank defines this dimensionless
radius
as the ratio
radius and the penetration
between
the bubble
depth, i.e. the thickness
Eng. Aspects X5 (1994)
is the relative bubble (s- ‘). By assuming radius
237-253
241
surface that
expansion
the increase
and the time intervals
rate
of
the
in the bubble
are very small, and
of the boundary layer. This represents the evolution of the bubble radius relative to the evolution of
the concentration profile is linear, Ax can be replaced in Fick’s first law in Eq. (8) by dxi and
the thickness of the boundary layer, and is a measure of the effect of time on bubble growth
the concentration difference dci calculated with use of Eq. (4) by dci in order to calculate the gas flux into the bubble
Ri s=(Dt)lil where D is the diffusion coefficient of the oversaturated gas in the liquid (m2 s-r) and t is time (s). Frank described the gas flux by
@FRANK
q27-rs3D3’2t”2 =
(6) Ai
where@FRANKis the gas flux according
to Frank’s theory (m s-l) and 4 is the concentration difference of CO2 in H,O between the bubble surface and the liquid bulk and is equal to dci (m” mP3). This model does not account for the expansion of the bubble surface and only pure diffusion is assumed, which is, as Frank has already mentioned, seldom completely justified. The transport of surface-active material to a steady-state expanding surface in a Langmuir trough by means of diffusion and convection is described by van Voorst Vader [ 141. Assuming that gas transport to the bubble behaves analogously by using Eq. (7) which is derived from the theory of van Voorst Vader, the ultimate diffusion distance under steady-state conditions can be calculated by assuming a constant concentration difference and a linear concentration profile of the oversaturated gas during the growth (AR) of the bubble. l/2
(7)
where Ax is the diffusion
distance
(m) and d In A/dt
where dci/dxi is the concentration gradient between the bubble surface and the liquid bulk at time t= i (m-l). An advantage of the theory van Voorst Vader is that it accounts for the expansion rate of the surface by means of the d In A/dt term which determines the contribution of convection of the total gas flux. However, it did not take into consideration the movement of the surface towards the liquid bulk, since the liquid surface in a Langmuir trough stays at a constant level. Furthermore, van Voorst Vader assumes a steadystate situation which prevents this theory being applied to the growing bubble, because during bubble growth the conditions, for instance the bubble surface expansion rate, change continuously. A new model was developed to describe bubble growth in terms of diffusion and “semi-convection” by considering the geometric changes that take place at the boundary layer during the growth of a bubble. In the model, Fick’s first law is applied. Several assumptions are made. (1) The concentration profile is linear as shown in Fig. 3. (2) The concentration difference between the bubble surface and the liquid bulk is not constant, but depends on the radius of the bubble as described in Eq. (5). (3) The model calculates the growth of a spherical bubble attached to a capillary, i.e. a cylindricalshaped cavity. In practice, however, the wall of the
--_ bubble
~boundary
liquid bulk
layer
(A A
‘liquid
bulk ~
(
)
Spherical shape
Bubble
~-
Capillary tube
I Distance [m]
Fig. 3. The assumed linear concentration profile in the hubble, at the bubble surface. in the boundary layer and in the liquid bulk.
I
Bubble
Neck’ shape
\
cavity at the rim is hydrophilic and at the moment that the bubble is detached from the capillary tip the bubble has a “neck” shape. as illustrated in Fig. 4. As will be discussed later. the bubble was assumed to be spherical. (4) Since the bubble is growing almost radially, it is assumed that there is no pressure gradient along the bubble surface that could result in mixing of the boundary layer with the liquid bulk that can be described as convection. Under these conditions, two geometric changes of the boundary layer occur during bubble growth, as presented schematically in Fig. 5. As illustrated in Fig. 5A, there will be an increase (drv,) in the thickness of the boundary layer in time, caused by the depletion of the gas that is diffusing into the bubble, resulting in penetration of the concentration gradient into the liquid. This increase can be described by the penetration theory presented in Eq. (9) and discussed by Lucassen [ 151 and Lucassen and van den Tempel [ 161.
where x is the distance
the front of the boundary
Fig.4. A schematic drawmg of the bubble model and as it is found in practice.
shape
used in the
layer penetrates into the liquid in time t (m), ‘7 is the viscosity of the liquid (N s m --‘) and /I is the density of the fluid (kg m-“). In Fig. 5B a decrease (d-x,) in the thickness of the boundary layer as a result of the stretching of the boundary layer caused by the surface expansion of the bubble is presented schematically. This is considered to be a “semi-convection term”, because it is not in agreement with the common definition of convection, i.e. an additional supply of solute gas perpendicular to and towards the bubble surface. Combining these two phenomena results in a model in which the bubble growth is described by diffusion and “semi-convection” as illustrated in Fig. 5C. Here the evolution of the thickness of the
C.G.J. Bisperink
and A. PrinslColloids
Surfaces A: Physicochem.
Eng. Aspects 85 (1994) 237-253
243
dx + dx,
dx
penetration theory
i
/
dx - dx,
expansion of surface
time _
dx,
dx,+, = dx, + dr,
dx.
interval i+l A.. ux,*,
Fig. 5. A schematic
representation
of the evolution
time
-
dx,,,=
of the thickness of the boundary see text.
boundary layer as a function of bubble growth is illustrated for two arbitrary time intervals i+ 1 and i + 2 during bubble growth. The boundary layer becomes thicker as a function of time as described by the penetration theory, and simultaneously becomes thinner due to the expansion of the bubble surface, i.e. stretching of the boundary layer. In order to perform the calculation of the growth rate of a bubble on a capillary or a cavity in a solid surface, the time interval (dt) needed for a very small increase of the bubble radius is used in this new theory. The volume of the boundary layer that is wrapped around the bubble can be calculated by considering the situation presented in
dx,,, + dx;
dx.
interval it2
layer during
bubble
growth.
For further
details,
Figs. 2 and 6. Here the concentration profile is assumed to remain linear during bubble growth. The starting point of the calculation is a half spherical bubble attached to the rim of the cylindrical “model” cavity for reasons explained later. By assuming the bubble growth during the time interval to be very small (i.e. AR -+ dR), the number of moles of CO,, indicated by 1\/Ii, that are present in the boundary layer with thickness Rhx at the beginning of the time interval i + 1 can be described by I%. (lo)
( 10) i=l
where
Vz,i is the volume
of the boundary
layer
C.G.J. Bisprrink and A. Prins/Colloids
244 Concentratb m’ m’s L-1
Swfaws
moles
3”
boundary layer
bubble
A: Ph!siuxhern.
of CO,
that
layer with thickness
liquid bulk
growth
Eng.
Aspects85 (1994) 237 253
are present
in the boundary
Ri;,;’ and available
AR is described
for bubble
by
M r+1_Mi_i+l=fAc,V~,,-i+l
I
/
’ liouid bulk
__
The number of moles of CO, indicated by MAR, necessary for the growth AR (i.e. representing a volume
increase
calculated
L
/ %x
with thickness dx (m3), Vt’q is the volume of the boundary layer calculated over previous time intervals (m3) and dci is the concentration difference in CO2 between the liquid bulk and the bubble surface (mol m3). In this equation also a correction was made for the bubble volume excluded by the capillary, by using the calculation method described in Eq. (3). The boundary layer with thickness Rzl contains a number of moles of CO, (Mi+r)
that can be calculated
Al/~_‘,+,
of the bubble)
can be
using
(14)
MV
Distance [m]
Fig. 6. A schematic drawing of the evolution of the conccntration profile during bubble growth. The number of moles of oversaturated gas available for bubble growth calculated by using Eqs. (9)-( 13) is shown as the shaded triangle.
(13)
Combining Eqs. (13) and (14) results in Eq. (15) with which the additional volume of liquid as described in Eq. ( 11) is calculated
2AVs-i+
L)
v*iq dx.c-ri
1
=
L
-I
(15)
M VAci
The total volume of the boundary layer that contains the number of moles of CO2 necessary and available for the growth AR of the bubble can now be calculated using
using
(11) i=l
is the additional volume of liquid where I/~$_,+i needed to reach a thickness Rzl of the boundary layer (m3). After the bubble radius has increased by AR (assuming a linear concentration profile) the number of moles of CO, indicated by JM,,~+ 1 that are still present in the boundary layer with thickness Rh!j’ can be described by Mi,i+l
izi C V~q + V~,i_i+l
=fAci (
By combining
i=1
Eqs. (11) and (12)
> the number
(12) of
j=i VI!4 (1-1+1)-
--
M VAc,
+ 1 I/t’4 i=l
(16) where V’i.q cc_r+l, is the volume of liquid containing the amount of CO, for the growth AT/;i,i+,, of the bubble (m” (liquid)). In this equation, the increase in thickness of the diffusion layer. assuming that the layer is not mixing with the bulk liquid, can be found from the summation of the volumes of the diffusion layers calculated over previous time
C.G.J. Bisperink
intervals.
and A. Prim/C&ids
The radius
Surfaces A: Physicochem.
of the bubble
plus the diffu-
sion layer (see Fig. 2) can be calculated 3 I& L
= 74
where
R,,
thickness
using l/3
2 + 3 co:I;i+1)cos3
pi) + R! 1
is the radius
of the bubble
of the diffusion
(17) plus
the
layer (m).
The derivation of Eq. ( 17) can be found in the appendix. The thickness dx of the diffusion layer at the beginning of the consecutive time intervals can be calculated using Eq. (18) Ax=Rdx-Ri
(18)
The concentration difference between the bubble surface and the liquid bulk can be calculated using Eq. (4) in which a correction has been made for the additional Laplace pressure in the bubble and the hydrostatic pressure acting on the bubble. The flux in the time interval i + 1 can now be calculated analogously to Fick’s first law with use of Eq. ( 19) by using the results of Eqs. (4) and ( 18)
(19)
where@ci+i+l) is the flux of gas in the time interval i + 1 (m s-l). Using the equation for the conservation of mass, the time Atci,i+l, needed for the bubble to grow AR in radius can be calculated using Eq. (20)
(20) where Ai is the surface area available for diffusion at time t = i (m2). It follows from the theory that (1) at high bubble growth rates (i.e. a large concentration gradient of the CO, between the bubble surface and the liquid bulk) the increase in the thickness of the boundary layer is considerably reduced due to surface expansion, i.e. thinning of the boundary layer as a result of stretching; and (2) at low bubble growth rates the CO, in the liquid bulk in the vicinity of the bubble is depleted, and the effect of stretching of
Eng. Aspects 85 (1994) 237-253
the boundary time available
layer is less because there is more for the penetration to proceed.
While the bubble grow until a certain bubble will detach bubble
245
is attached to a cavity, it will bubble size is reached and the from this cavity. This “final
size” depends
between According
the to
on the balance
buoyancy
and
Ronteltap
of the forces
the surface
[lo],
the
tension.
moment
of
detachment is reached when the buoyancy force becomes bigger than the adhesive force (i.e. the vertical component of the surface tension of the bubble), as shown schematically in Fig. 7 and in the equation VApg = ydyn sin (a)0
(21)
where Ap is the density difference between the gas and the liquid (kg mP3), V is the volume of the bubble (m3), 0 is the perimeter of the bubble where it is attached to the cavity (m), yd,,,, is the dynamic surface tension of the expanding bubble surface and a is the apparent contact angle (N m-l), between the liquid and the solid phase (degrees). According to Morra et al. [ 171 and a review by Good [ 181, the wetting properties of the glass (i.e. the contact angle between the bubble and the glass wall) are important parameters. It can be predicted that when, for instance, the glass is hydrophobic,
I
i
‘\
BUBBLE
\
I
i,
Fig. 7. A schematic representation bubble attached to the cavity.
/
of the forces acting
on the
C.G.J. Bispmnk
246
and A. Prins/Colloidc
Surfaces A: Physicochem. Eq. Aspects 85 (1994) 237-253
a gas bubble growing out of a cavity would spread over the glass surface and, furthermore, the contact
To study bubble formation in carbonated liquids in the absence of liquid flow (motion), an experi-
angle
mental
method
bubble known
growth in a controlled way under wellconditions. From the theories regarding
whether
or not a gas pocket can exist, as described
between
decrease if the bubble surface
the
bubble
and
the glass
would
dynamic surface tension of the is higher, as follows from the
equation ;!SG
-
ys,_
=
;‘LG
cos(x)
by Ward (22)
where ySG is the surface tension of the solid-gas surface (N m-l), ys,, is the surface tension of the solid-liquid interface (N m-‘), yLG is the surface tension of the liquid-gas surface (N m-l), and x is the contact angle of the liquid with the solid phase (degrees). The bubble radius at the moment of detachment will, consequently, be large in the case of hydrophobic surfaces as a result of the large value of x Since, in reality, close to detachment a bubble shape is observed as shown in Fig. 4B, sin(a) = 1 and Eq. (21) can be rewritten as VApg = ydyn0
(23)
Ronteltap [lo] concluded that the bubble size is mainly determined by the surface tension at the moment of detachment. Because nucleation and growth of a bubble is a very rapid process, the bubble surface expansion rate is high. Therefore, according to Ronteltap, the increasing surface tension due to surface expansion can be significant. This surface tension depends on the history of the bubble surface and the surface dilational viscosity at the given expansion rate. The oversaturation value of the gas determines, amongst others, the expansion rate. From this phenomenon it can be predicted that a low dynamic surface tension during expansion results in smaller bubbles. Furthermore, during the dispensing of carbonated beverages, the liquid flow in the tap causes an earlier detachment of the bubbles: the liquid flow strips off the bubbles from the cavity. Consequently MahC et al. [ 191 defined detachment as an equilibrium between shear rate forces and surface tension forces. A large shear rate results in a higher growth rate but also in an earlier detachment of the bubble from the cavity. Therefore, the bubbles will remain smaller.
was
developed
et al. [2,11],
Ward
by us to allow
and
Levart
[ 121,
Peters [ZO] and Stoopen
[21], a model cavity was
developed
of a stable gas pocket at
which consists
the required
conditions.
Experimental The “model” cavity was made from a capillary tube (Hirschmann, glass with an inner diameter of 1 mm) which was melted and elongated into a cylindrical tip with an inner diameter varying from 40 to 200 pm. The end of the tip was cut off sharply, resulting in a geometry as presented in Fig. 8. The bottom of the cavity was formed by means of glue (Bison Combi-Rapide) that was brought into the capillary from the opposite side of the cavity being prepared. Any release of surfaceactive components from the glue was checked by half bubble
,_-,-
,I’
‘-
glue
‘dead’ volume = half bubble volume
,,r
’
,.’
Capillary tube
Fig. 8. A schematlc this study.
presentation
of the model
catty
used in
C.G.J. Bisperink and A. PrinslColloids
measuring
the equilibrium
Swfaces
surface
water in the vessel containing of the used
capillaries
equilibrium
surface
tension
of the
the capillary.
None
was found
tension
A: Physicochern.
to lower
of pure
bubble
with a diameter the cavity
(i.e. was of a
equal to the tip diameter.
In this way, the possible influence inside
247
the
water
72 mN mm’). The volume of the cavity adjusted to approximately half the volume
Eng. Aspects 85 (1994) 237-253
on the bubble
of the gas volume growth
rate
as
described by Zuidberg 1221 was made negligible. During bubble growth, the bubble pressure decreases due to the decreasing Laplace pressure. The result is an expansion of the total gas volume leading to an increased bubble growth rate. If the gas volume within the cavity is above a threshold volume, the bubble will be blown off the cavity as soon as it becomes larger than half the bubble, i.e. the maximum Laplace pressure. Gas volumes smaller than the threshold value will contribute to the bubble growth rate but will not necessarily result in the bubble being blown off the cavity. The contribution to the bubble growth will decrease on decreasing the gas volume in the cavity. Furthermore, after a bubble is released from the cavity it is assumed that a half sphere remains at the cavity, as depicted in Fig. 8. This half sphere is the start condition of the calculation method described above. During experiments, the capillary tube was placed vertically in a temperatureand pressurecontrolled Perspex vessel, as shown schematically in Fig. 9, simulating the cavity present in the bottom of a glass containing a carbonated liquid. The vessel was filled with 800 ml of a buffer solution prepared with sodium acetate/acetic acid at pH 4. The solution was saturated with carbon dioxide at atmospheric pressure by constant flushing of the headspace in the vessel with the gas (i.e. C02) while stirring the liquid. The degree of saturation was monitored qualitatively by using COzsensitive electrode (Orion, model 95-02) connected to a pH/pX/millivoltmeter (Ankersmit, model A161), and saturation was assumed when the readout was at a constant level for 10 min. The saturation concentration of CO, in the buffer at
CAMERA LIGHT DIFFUSER INTERVAL
TIMER
Fig. 9. A schematic representation study to measure the bubble growth
of the device used in this rate in a carbonized liquid.
1 atm was calculated using the tabulated values of the saturation concentration of CO, in water at 1 atm and 20°C [23]. A correction was made for pH 4 by using the equilibrium constant for the dissociation of dihydrocarbonate in water. After the saturation with CO, was established, the pressure was reduced to the pressure required during measurement, with use of a vacuum pump (KNF Neuberger, miniport N79 KN.18), down to a minimum of 0.33 atm. The pressure was measured using a U-tube manometer filled with mercury. The resulting oversaturation of the carbon dioxide causes growth of the gas pocket into a bubble by the transport of gas. The bubble growth rate at the tip was monitored by taking photographs using a photo camera (Nikon F801) and a flash light (Metz 60 CTl) at consecutive time intervals up to a maximum speed of three photographs per second. The frequency was monitored using an interval timer with an accuracy of 0.1 s. To obtain photographs of good quality, external light disturbances were eliminated by covering the vessel with black paper except for the rear wall, and leaving a small window in the front for taking the photographs. By diffusing the light from the
saturation
of a sodium
acetate/acetic
acid buffer
flash light by placing an optical diffuser between the light and the vessel, as depicted in Fig. 9, exposures of good quality were obtained. The photographs were evaluated using an image
solution (pH 4) with CO, followed by a pressure decrease resulting in oversaturation of the gas. The results presented in Fig. 10 are obtained from triple
analyser
measurements
with a standard
The
growth
(Joyce-Loebl,
Magiscan)
with which
the
bubble
rates
error were
of f2.5
urn.
measured
at
bubble volume, area and radius were determined. By using these data, it is also possible to calculate
0.36 atm
the surface expansion The expansion rate
rate of the growing bubble. can be reproduced in a
80.3 urn and a COz concentration in the liquid bulk of 0.031 mol 1-l. This corresponds to an
Langmuir tough equipped with a caterpillar belt with barriers as described by Prins 1241, or in the overflowing cylinder as described by Padday [ 251, Piccardi and Feronni [26,27]. Joos and de Keyzer [28], Bergink-Martens et al. [29] and BerginkMartens [30]. With this information, the rheological behaviour of the bubble surface can be described by the surface dilational viscosity as a function of the expansion rate of the liquid surface. The developed model was used as a computer program written in Asyst 4.0 (Keithly) on a computer (Estate 80486133 mHz) to predict the bubble growth rate. By applying very small values of AR (in our case, 5 x lo-” m) using the computer program, the bubble growth rate was calculated. The theories of van Voorst Vader and Frank were tested by using the same values of AR as in the new theory presented in this paper in order to calculate the time interval (At) or flux (@) needed for the bubble growth (AR). It is not possible to predict a relative surface expansion rate (d In A/dt)
oversaturation value of the CO, at 0.36 atm of 3.57. During the experiments, the bubble radius increased during bubble growth from 80.3 urn up to a final maximum bubble radius at the moment of detachment. of 960 urn. Due to the decrease in Laplace pressure between the start of bubble growth and the moment of detachment. the concentration difference calculated using Eq. (4) increases to 5.7 x 10e5 mol I- ‘. Therefore, the concentration difference (AC,) of CO, was assumed to be constant at a level of 0.0223 mol l- ‘. Nevertheless, at very small radii of the cavity (i.e. the initial bubble size) and applied pressures close to 0 atm, the concentration at the bubble surface at the beginning of bubble growth will be considerably higher due to the higher Laplace pressure. This results in a smaller concentration difference (Aci) and the bubble growth rate will be consequently slower. The relatively slow growth rate at the start of
during bubble growth because it is determined by the growth rate itself. Therefore, the flux according to the theory of van Voorst Vader was derived using the values of AR and the values of At calculated with the developed theory. In this way, the ultimate diffusion distance indicated by AX could be calculated using Eq. (7). This would give an indication of the differences between the fluxes calculated according to the theories of Frank and van Voorst Vader and the present theory. Results and discussion As mentioned earlier, measurements of growth rates of the bubbles were performed
the by
using
a model
cavity
with a radius
bubble growth, seen in Fig. 10. is the result combination of the higher Laplace pressure,
of
of a but,
more importantly, as a result of the relatively large part of the bubble surface that is excluded by the model cavity (i.e. the capillary) and therefore not available for gas transfer into the bubble. The surface to volume ratio becomes more favourable for gas transfer when the bubble becomes larger. because the excluded surface and volume segment become relatively small. By assuming the bubbles to be spherical instead of the situation found in practice, where an almost spherical bubble is attached to the capillary with a “neck” as shown in Fig. 4B, the maximum error during the experiments presented here was a 3% underestimation of the bubble volume, which is. in our opinion,
an acceptable
deviation
from reality
C.G.J. Bisperink
and A. PrinslColloids
Surfuces A: Physicochem.
Eng. Aspects 85 (1994) 237-2.53
249
he f4
1000 100, 10: 1: 0.1 E 0.01: 0.001: 0.0001 -0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
radius lmm]
Fig. 10. The time necessary for the bubble to reach a certain radius (the y axis represents the time, and the radius is plotted on the x axis). The experimental data are plotted as filled triangles. The bubble growth rates calculated with the three different theories are plotted as solid lines. The liquid used and the conditions applied are given in the text.
that can be neglected. Furthermore, the maximum error is made for the “final” bubble size just before detachment from the cavity. At smaller bubble sizes, the error was smaller and consequently the error in the calculation of the gas flux into the bubble, i.e. the growth rate, was reduced. The theories of Frank and van Voorst Vader underestimate the growth rate of the bubble as can be seen in Fig. 10. The theory of van Voorst Vader calculates the maximum diffusion distance at steady-state
conditions
of the surface
expansion
rate. However, during bubble growth, steady-state conditions are not established. Besides diffusion, van Voorst Vader describes an additional convective diffusion caused by a concentration gradient existing in the thin subsurface layer in that region of the liquid surface in a Langmuir trough where the surface expansion was applied. Analogously, a concentration gradient can be found in a thin layer of liquid, i.e. a boundary layer around the growing bubble. Nevertheless, the bubble growth rate cannot be predicted using the theory of van Voorst
Vader since the relative surface expansion rate cannot be predicted, and the latter determines, with the calculation of the maximum diffusion distance using Eq. (7), the growth rate of the bubble. Because the relative surface expansion rate (d In A/dt) cannot be predicted for a growing bubble, d In A/dt was calculated with the data obtained by using the developed theory presented in this paper. In this way, an averaged d In A/dt during a time interval was used in Eq. (7) in order to calculate dx by assuming steady-state conditions during that period, which is not completely justified. i\ievertheless, since the time intervals are small, the error made by assuming the steady state is minimized. Frank overestimates the bubble growth rate when the bubble is still small. This is due to the fact that Frank’s model calculates the growth rate of a full sphere. In our experimental set-up, a part of the bubble up to a maximum of a half sphere at the start of bubble growth is excluded by the capillary. Since the diffusion of gas into the bubble
depends on the available surface area for gas transfer, Frank’s theory operates with a surface area that is too large compared with the actual
pressure was decreased experimentally by adding Teepol to the earlier-mentioned buffer solution. In
available surface area. When Frank’s theory would be adapted to the present conditions. i.e. taking
the developed theory are depicted for a buffer solution with a dynamic surface tension in expansion of 72 mN m ’ and for a 1% (v/v) Teepol in
Fig. 11, the results and the curves calculated
into account an excluded bubble volume segment, the bubble growth rate would be underestimated
buffer solution,
also at small
bubble
with
bubble
In the latter
radii.
radii,
as is found case,
Frank’s
at larger theory
which is above the CMC of Teepol.
a dynamic
27 mN m-l.
using
surface
The radius
tension
in expansion
of the model
cavity
of used
underestimated the bubble growth rate because it did not take into consideration the stretching of the boundary layer due to expansion of the bubble surface. Although the stretching of the boundary layer occurs from the beginning to the detachment of the bubble from the cavity, the effect of the actual surface area (i.e. corrected for the excluded bubble volume segment) and the used surface area (i.e. the surface area of a spherical bubble as used by Frank) compared to the high extent of stretching at the beginning of bubble growth, plays a more important role when the bubbles are small. At larger bubble radii. the surface area excluded by the capillary becomes relatively smaller and in the end even negligible, and the effect of the thinning of the boundary layer caused by stretching becomes more pronounced, although the magnitude of the stretching decreases due to the decreasing bubble growth rate (dR/dt) as a result of the decreasing value of the surface to volume ratio. Therefore, at larger bubble radii during growth. Frank’s theory predicts a smaller gas flux into the bubble because the diffusion distance calculated with his theory,
was 58 pm and for both experiments the CO, concentration was adjusted to 0.035 mol 1-l at 1 atm and the applied pressure during bubble growth was 0.36 atm. In the figure it can be seen that the lower surface tension has no significant effect on the bubble growth rate. Compared with the applied pressure, the additional Laplace pressure is negligible considering the bubble radii found in these experiments. For radii that are much smaller and when much lower pressures are used, the correction for the Laplace pressure becomes more important. The present theory describes the bubble growth rate rather well. It can be concluded that with simple considerations regarding the geometric changes that take place at the bubble surface and consequently the thickness of the boundary layer, the bubble growth rate can be calculated using Fick’s first law when convection is not involved. In practice, these circumstances are assumed to be present when bubbles grow at cavities at the horizontal bottom of a vessel containing an oversatu-
which is analogous to the penetration depth mentioned in Eq. (9), is too large. In the theory presented here, the history of the bubble is an important parameter. The evolution of the boundary layer in time depends on the bubble growth rate, but also on the starting point of the calculation, which is, in our case, a half sphere with a radius equal to the cavity radius. The evolution of the thickness of the boundary layer depends also on the Laplace pressure of the bubble. A higher Laplace pressure results in a smaller concentration difference between the bubble surface and the liquid bulk. The Laplace
cavities on tilted walls need an adaptation of this theory. Not only do these bubbles deviate more from the spherical shape. but the angle of attachment is not uniform. In addition during expansion of the bubble surface the shape of the bubble deviates even more from the spherical one which is assumed in our theory. In Figs. 10 and 11 it can be observed that the model calculates an earlier moment of detachment using Eq. (22) compared with the measured maximum bubble size. This could be the result of an error made in the measurement of the perimeter (0) and/or the contact angle (2) and./or the surface
rated
carbonated
liquid.
Bubbles
growing
at
C.G.J. Bisperink
and A. PrinslCoNoids
Surfaces A: Physicochem.
Eng. Aspects 85 (1994) 237-253
251
time [s]
3s
2520-
half sphere
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
radius [mm] Fig. 11. The time necessary for the bubble to reach a certain radius (the JJ axis represents the time and the radius is plotted on the x axis). The experimental data for a buffer solution with a dynamic surface tension of 72 mN mm 1 are plotted as filled circles and the experimental data for a buffer solution containing 1% (v/v) Teepol with a dynamic surface tension of 27 mN m-r are plotted as filled stars. The fit curves calculated using the developed theory are presented as solid lines.
tension in expansion (ydyn) at the moment of detachment. The rim of the model cavity is assumed to be smooth and circular. In reality, it is possible that the perimeter is larger due to irregularities of the rim that are not observed on using the microscope for measurements of the cavity’s radius. Furthermore, it could be possible that a minor error is made by assuming the cavity to be circular, and in reality it deviates from circularity. The apparent contact angle (c() may differ due to poor wetting properties of the liquid with the surface in which the cavity is present. This was, however, not observed on the photographs taken during the experiments. The surface tension in expansion may have been smaller than was assumed (i.e. 72 mN m-l). Although this could be a possibility, it is not very likely since the equilibrium surface tension of the buffer solution present in the experimental set-up was found to be 70.5 mN m-’ before and after the experiment. Furthermore, the bubble surface is expanded during bubble growth, and the dynamic surface
tension in expansion has to be used; this was found to be 72 mN m-’ measured in the overflowing cylinder and the Langmuir trough equipped with a caterpillar belt at relative surface expansion rates between 0.1 and 1 s- ‘. Evaluating the possibilities mentioned here, the most probable error is caused by an incorrect measurement of the perimeter where the bubble is attached to the cavity. A lower surface tension has a considerable effect on the bubble size at the moment of detachment from the cavity. As can be seen in Eq. (23), the final bubble volume at the moment of detachment is proportional to the dynamic surface tension in expansion at the moment of detachment. This effect is illustrated in Fig. 11, where the two liquids have a dynamic surface tension in expansion of 72 Furthermore, the and 27 mN m-‘, respectively. bubble volume is also proportional to the radius of the cavity. This effect can be found by comparing the final bubble sizes found in the buffer solutions equal dynamic surface tensions (i.e. with 72 mN m-‘) but formed at cavities with different
C.G.J. Bisperink
252
and A. PrinsiColloids
radii of 80.3 urn and 58 urn, shown in Figs. 10 and 11, respectively. In the case of a bubble that is not attached to a cavity but is present in the liquid bulk, this theory and the calculation describe the growth
method can also be used to of this bubble. For this reason,
Eqs. (3) and (17) have to be rewritten
3 I/“:4 R,x
=
(‘-‘+‘)
471
1
more simply:
R?
II
(25) 15 16
The advantage of the theory presented here is that it can be used as a simple computer program with which the bubble growth rate can be calculated. A disadvantage is that it is only valid for describing bubble growth in systems without convection. For instance, stirring of the liquid would result in an increased gas supply towards the bubble and therefore in a higher growth rate. Furthermore, the starting point of calculation is important. The calculation method cannot be used for the calculation of the bubble growth started some time after the beginning of growth since the evolution of the thickness of the boundary layer depends on time. Finally, the moment of detachment depends on the perimeter of attachment of the bubble to the cavity, the wetting properties of the liquid and the cavity and the dynamic surface tension of the liquid. Therefore. these parameters have to be known in order to be able to calculate the “final bubble size”.
17 18 19 20 21 22 23 24
25 26 27 28 29 30
Acknowledgement Part of this research is supported within the “Bufom” project (EU-267).
10
12 13 14
l/3
+
9
by Eureka
References M. Blander, Adv. Colloid Interface Sci., 10 (1979) 1. C.A. Ward, W.R. Johnson, R.D. Venter, S. Ho, T.W. Forest and W.D. Fraser, J. Appl. Phys., 54(4) (1983) 1833. R.H.S. Winterton, J. Phys. D, 10 (1977) 2041. P.M. Wilt, J. Colloid Interface Sci.. 112(2) (1986) 530. P.A. Ciholas and P.M. Wilt. J. Colloid Interface Sci., 123( 1) (1988) 296.
Swfi~es
A: Physicochem.
En‘?. Aspects 85
( 1994) -737~-253
R. Cole, Adv. Heat Transfer. 10 (1974) 85. L.N. Tao, J. Chem. Phys., 69(9) (1978) 4189. S. Lubetkin and M. Blackwell, J. Colloid Interface Sci.. 26(2)(1988)610. P.S. Epstein and MS. Plesset, J. Chem. Phys.. 18(11) (1950) 1505. A.D. Ronteltap, Ph.D. Thesis. Agricultural University, Wageningen, 1989, p. 133. C.A. Ward, A. Balakrishan and F.C. Hooper, J. Basic Eng., 92 (1970) 695. C.A. Ward and E. Levart. J. Appl. Phys., 56( 2) ( 1984) 491. F.C. Frank, Proc. R. Sot. London, 201A ( 1950) 586. F. van Voorst Vader. F. Erkens and M. van den Tempel. Trans. Faraday Sot., 60 ( 1964) 1170. J. Lucassen, Trans. Faraday Sot.. 64 ( 1968) 222 I J. Lucassen and M. van den Tempel, Chem. Eng. Sci.. 27 (1972) 1283. M. Morra, E. Occheillo and F. Garbassi, Adv. Colloid Interface Sci., 32 (1990) 79. R.J. Good, J. Adhesion Sci. Technol., 6( 12) (1992) 1269. M. Maht. M. Vignes-Adler and P.M. Adler. J. Colloid Interface Sci.. 126( 1) ( 1988) 329. B. Peters, M.Sc. Thesis. Agricultural University, Wageningen. 1990. p, 47. L. Stoopen, M.Sc. Thesis. Agricultural University, Wageningen, 1993, p. 38. A.F. Zuidberg, Internal Report, Agricultural University. Wageningen, 1992. R.H. Perry and D. Green, Perry’s Chemical Engineers Handbook. McGraw-Hill Book Co.. Singapore. 1984. A. Prins. Dynamic Surface Properties and Foaming Behaviour of Aqueous Surfactant Solutions. in R.J. Akers (Ed.). Foams, Academic Press. London, 1976, p. 51. J.F. Padday, Proc. Int. Congr. Surf. Act.. 2nd London, 1 (1957) I. G. Piccardi and E. Feronni. Ann. Chim. (Rome), 41 (1951) 3. G. Piccardi and E. Feronni, Ann. Chim. (Rome). 43 (1953) 328. P. Joos and P. De Keyzer, Europhys. Conf. Abstr., 3F (1980) 156. D.J.M. Bergink-Martens, H.J. Bos, A. Prins and B.C. Schulte. J. Colloid Interface Sci.. 138 (1990) 1. D.J.M. Bergink-Martens. Ph.D. Thesis, Argricultural University, Wagenmgen. 1993, p. 151.
Appendix A The derivation of the volume time t = i is described by
of the bubble
at
v, = ($R!)- [$(Ri-R,cos pi)'
x(3Ri-Ri+ Rices/&)I =($zR;)-[C~TCR~(~3 cos ai+cos3 = $cR?(2 + 3 cos pi - cos3 Bi)
/I)] (Al)
C.G.J. Bisperink
The volume described E+r
and A. PrinslColloids
of the bubble
analogously
Surfaces A: Phgsicochem.
at time t = i + 1 can be
to Eq. (Al)
=$LR;+~(2 + 3 COs pi+r --OS3 fli+r)
As can
be seen in Fig. 2, the spherical
(A2)
Eng. Aspects 85 (1994) 237-253
used to calculate R,,using Eq. (17). The derivation of Eq. (17) is described in Eq. (A5): v&r (t+++l)
volume
=
v,x-
=
37r[R&(2 + 3 cos pi - cos3 pi)
representing the bubble (Ri) and the diffusion layer (Rdx) have the same centre, therefore Eq. (A3) is =
R,,COSBdx
(A3)
Consequently, the spherical volume V,, with radius R,,(Eq. (A4)) can be derived from Eq. (Al) v,, = frrR;,( 2 + 3 cos pi - cos3 pi) The volume
of liquid
calculated
v
- R;( 2 + 3 cos pi - cos3 ji)]
valid Ri COSpi =
$c(Rj, - R?)(2 + 3 cos pi - cos3 pi) v!iq t-1+1
&(R&RRf)=
2+ 3
COS
pi
-
COS3
pi
3 v!‘q
R;,=
7r(2 + 3 CSii”
(A4) with Eq. (16) is
253
co? Pi) + R!
=
n(2 + 3
CO$~i”
1 l/3
3 v!i4 &x
COS3
Bi)
+
R3
(A5)
