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Colloids and Surfaces A: Physicochemical andEngineering Aspects
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ree drainage of aqueous foams stabilized by mixtures of a non-ionic (C12DMPO)nd an ionic (C12TAB) surfactant
nda Careya, Cosima Stubenraucha,b,∗
Universität Stuttgart, Institut für Physikalische Chemie, Pfaffenwaldring 55, 70569 Stuttgart, GermanyUniversity College Dublin, School of Chemical and Bioprocess Engineering, Belfield, Dublin 4, Ireland
i g h l i g h t s
Influence of the composition of asurfactant mixture, the bubble sizeand the initial liquid fraction on freedrainage.Immobile surfaces and thusPoiseuille-like flow under allexperimental conditions.Drainage behavior is not correlatedwith respective foam stabilities.
g r a p h i c a l a b s t r a c t
r t i c l e i n f o
rticle history:eceived 18 September 2012eceived in revised form3 November 2012ccepted 15 November 2012vailable online 29 November 2012
eywords:
a b s t r a c t
The study at hand investigates the influence of the composition of a surfactant mixture on the freedrainage of the respective foams. The Foam Conductivity Apparatus (FCA) was used to study free drainageof foams of homogeneous initial liquid fraction εinit. The foams were stabilized with mixtures of thenon-ionic surfactant dodecyldimethyl phosphineoxide (C12DMPO) and the cationic surfactant dodecyltrimethylammonium bromide (C12TAB) at mixing ratios of C12DMPO:C12TAB = 1:0, 50:1, 1:1, 1:50 and 0:1.In all cases the liquid fraction ε follows a power law with respect to time t, i.e. ε ∼ t� with −1.0 < � < −0.8.The �-values are all very similar which indicates a Poiseuille-like flow (−1 < � < −2/3) for all samples
irrespective of changes in composition and bubble size. For all studied mixtures drainage led to fairly dryfoams (ε ∼ 10−3), which then either coalesce (C12DMPO and 50:1 mixture) or drain further (1:1 mixture).An evaluation of the critical liquid fraction εcrit (liquid fraction at which foam coalescence sets in abruptly)revealed that εcrit is independent of the foam height and the bubble size which is in line with literature.However, it depends on the composition of the surfactant mixture and decreases with increasing contentof C TAB.
urfactant mixtures12
. Introduction
In order to measure the drainage of aqueous foams one needs
o determine the time evolution of the liquid fraction ε. The latters typically calculated from the foam’s conductivity. Although theonductivity of aqueous foams has been investigated in numerous
experimental studies [1–6], conductivity measurements still needto be optimized [7,8]. Recently, Karapantsios et al. examined criti-cal design aspects for measuring the liquid fraction ε in foams suchas electrode size, shape, separation distance, intrusiveness, excita-tion current frequency and multiplexing of electrical conductanceprobes [7]. Theoretical considerations of how to calculate the liq-uid fraction ε from the conductivity of a foam has made significantstrides. Lemlich developed the relationship between the foam con-
ductivity and its liquid fraction ε for low liquid fractions (ε → 0)through theoretical analysis [9]. For high liquid fractions (ε → 1)Lemlich [10] suggests to use the Maxwell formula [11]. Recently,Feitosa et al. successfully used an empirical relationship based on a
arge amount of current and historical experimental data to calcu-ate liquid fractions from conductivity data over the entire ε range12,13].
The exponent � of power law fits to drainage curves (liquidraction ε plotted as a function of time t) allows characterizinghe surface mobility of the foam during drainage [14]. The surface
obility depends on the bubble size as well as on the interfacialnd bulk properties [14,15]. Indeed models predict that the surfaceobility is inversely proportional to the bubble size [16]. In case
f a high surface mobility the surfaces are fluid and mobile and alug-like flow of the liquid through the Plateau Borders is observed17–20]. In the case of a low surface mobility the surfaces are rigidnd immobile which results in a Poiseuille-like flow [16,17,20–22]nd thus a slower drainage rate. The drainage regime depends onhe type and concentration of the surfactant [23]. Another param-ter of interest which is obtained via free drainage measurementss the critical liquid fraction εcrit, i.e. the liquid fraction at whichhe foam collapses[1], which, in turn, is an important parameteror foam stability [24].
Free drainage of aqueous foams stabilized by the non-ionic sur-actant dodecyldimethyl phosphineoxide (C12DMPO), the cationicodecyl trimethylammonium bromide (C12TAB), and their mix-ures at mixing ratios of C12DMPO:C12TAB = 1:0, 50:1, 1:1, 1:50,nd 0:1 are investigated in the present work. This work fol-ows a line of studies on this particular mixture in which theirespective water–air surfaces, foam films and foaming behaviourave been investigated [25,26]. To truly characterise foams, it
s necessary to also examine the individual components of theoam structure, namely the surfaces and the foam films. This haseen demonstrated recently in an extensive study of mixtures ofwo non-ionic surfactants, namely hexaoxyethylene dodecyl etherC12E6) and n-dodecyl-�-D-maltoside (�-C12G2) [27]. Here withinor the C12DMPO:C12TAB mixtures adsorption at the water–airurface was studied via surface tension measurements, while theroperties of horizontal foam films of thickness h < 100 nm werexamined via a thin film pressure balance [25]. Foaming behaviour
both foamability and foam stability – was studied by the com-ercially available FoamScan method [26]. However, the FoamScanethod did not allow for the study of foams with a uniform liquid
raction. This is exactly where the present work jumps in: foamsith uniform liquid fractions were generated and their drainage
ehaviour was studied. To get a clear picture we studied drainages a function of the bubble size, the initial liquid fraction, the com-osition and the total surfactant concentration.
. Experimental
.1. Materials
The non-ionic surfactant dodecyldimethyl phosphineoxideC12DMPO) was synthesised and purified as described elsewhere28]. The cationic surfactant dodecyl trimethylammonium bromideC12TAB) (purity ∼ 99%) was purchased from Aldrich and purifiedy three-fold recrystallizing from pure acetone to which tracesf ethanol were added. Perfluorohexane (purity, 99%) was pur-hased from Aldrich and used as received. Sodium chloride (NaCl)as obtained from Merck and roasted at 500 ◦C over night to drive
ff organic contaminants. The solutions were prepared with Milli-® water. All glassware was cleaned with deconex® UNIVERSAL 11
rom Borer Chemie and rinsed thoroughly with water before use.
.2. Foam conductivity apparatus
.2.1. Experimental setupA very accurate technique for foam measurements is the
oam Conductivity Apparatus (FCA) [1–3]. In the FCA, a foam
: Physicochem. Eng. Aspects 419 (2013) 7– 14
of homogeneous bubble size is generated by bubbling nitrogen(N2) gas saturated with perfluorohexane (C6F14) through a per-forated disc into the surfactant solution which is placed inside aPlexiglas column (square cross-section = 25 mm × 25 mm, lengthof column = 650 mm). Three perforated discs with evenly spacedholes of varying sizes were used, namely a disc with 1 hole of1 mm, one with 31 holes of 0.3 mm and one with 31 holes of0.04 mm in diameter. A gas flowrate range of 0–200 ml min−1 wasused. Perfluorohexane was used to slow down coarsening as thefluorinated gas is insoluble in water [14,29,30].
During foam generation drainage takes place due to gravity andcapillarity so that one usually ends up with an inhomogeneous foam– the foam is dry at the top and wet at the bottom. With the FCA,however, the foam can be wetted from the top with the surfactantsolution via a peristaltic pump. The added solution compensatesfor the solution lost due to drainage and leads to a foam with auniform liquid fraction from the top to the bottom. Along the col-umn 10 pairs (1 for liquid and 9 for foam measurements) of nickelplated brass electrodes are located and connected to an LCR meter.Once a foam with a homogeneous liquid fraction is formed, thegas flow and the peristaltic pump are switched off and the elec-tric conductivity is measured as a function of time t every 10 s atall electrodes simultaneously. Liquid conductivity measurementswere performed in an alternating electrical field (AC). To over-come polarization which occurs with highly conducting liquids onerequires high frequencies of up to 4 kHz [6,31].
The foam first dries at the top of the column and a dry frontpropagates down through the foam, with the liquid accumulat-ing at the bottom. At the end of the experimental runs a residualheight of bubbles persisted in the lower part of the column. Thisfoam is stable and no more evolves or coalesces as a balancebetween drainage and hydrostatic capillary pressures within thefoam is reached. Visual observations during drainage suggest thatcoalescence events occur only at the top of the foam. This wasexperimentally checked by rewetting the foam from the top withthe flow rate used for preparation of the sample and remeasuringthe liquid fraction. Comparing this to the liquid fraction measuredduring the preparation of the sample, it was concluded that thenumber of bubbles were unchanged before the arrival of the rup-ture front [1,32]. This was also confirmed using image analysis (seeSection 2.2.2) before drainage and after rewetting of the foam.
2.2.2. Determination of bubble sizeStructural information is of utmost importance when examin-
ing foam properties. FCA measurements allow for determining thebubble size of the generated foam. Two techniques can be used todetermine the bubble size, namely crystallizing the monodispersefoam in a capillary tube of known dimensions and/or image anal-ysis of the plateau borders along the border of the column [2,13].For foams stabilized by C12DMPO, C12TAB and their respective mix-tures crystallizing monodisperse foams in a capillary tube was notpossible due to their instabilities. Thus, throughout this study imageanalysis of the plateau borders at the wall of the column was usedto measure the bubble size. The relationship between the surfaceand the internal structure of a foam was investigated both exper-imentally [33,34] and via simulations [35,36]. Briefly, a picture ofthe foam column is taken (see Fig. 1) and the average length ofthe parietal (parietal = of or relating to the walls of a part or cavity)plateau borders LPPB is measured. An average of 10 bubbles is rea-sonable if the foam is almost monodisperse otherwise a statistical
analysis over 50 bubbles should be carried out. The average lengthof a plateau border of a bubble located in the interior of the foamLPB can be calculated from the average length of the parietal plateauborders LPPB [37]. For size dispersions (defined as the ratio of the
E. Carey, C. Stubenrauch / Colloids and Surfaces A: Physicochem. Eng. Aspects 419 (2013) 7– 14 9
F CA) cai averagf h LPB ∼
sd
L
rdbwLa
2
bcdf�
ε
dlestcogrEaεic
fltohe
ig. 1. The structure of foams generated with the Foam Conductivity Apparatus (Fmage analysis along the side of the foam column. Experimentally one obtains the
or LPB to be calculated. (left) Wet foam with LPB ∼ 0.3 mm and (right) dry foam wit
tandard deviation over the average) of the order of 30% one caneduce [2]
PB = LPPB
1.2. (1)
The bubble size can be varied by using different perfo-ated or porous discs (see Section 2.2.1). Foams with threeifferent bubble sizes were produced, namely large bub-les with LPB = 2.3 − 2.6 mm (std = 0.5 mm), medium bubblesith LPB = 1.0 − 1.2 mm (std = 0.3 mm) and small bubbles with
PB = 0.25 − 0.3 mm (std = 0.05 mm), where std is the standard devi-tion over 10 bubbles.
.2.3. Liquid fraction evaluationThe FCA allows for the variation of the initial liquid content εinit
y adjusting the flow of the foam solution on to the top of the foamolumn prior to drainage measurements. An empirical relationshipeveloped by Feitosa et al. was used to calculate the liquid fraction εrom the conductivity data obtained [12]. The (local) liquid fraction
is expressed in terms of the relative electrical conductivity by
= 3�el1 + 11�el
1 + 25�el + 10�2el
. (2)
The relative electrical conductivity �el can be defined as the con-uctivity of the foam �foam relative to the conductivity of the bulk
iquid �liquid, i.e. �el = �foam/�liquid. Eq. (2) is used throughout tovaluate the liquid fraction ε. The initial liquid fraction εinit corre-ponds to the constant liquid fraction at t = 0 which correspondso the time at which the liquid into the system from the top of theolumn is stopped. As the figures given throughout are representedn log–log scales the εinit-values are not shown graphically but areiven in the respective tables. The critical liquid fraction εcrit cor-esponds to the lowest liquid fraction observed prior to rupture.rror bars used for εcrit throughout correspond to measurementst all electrodes and thus over the entire foam column. Measuring(t)-curves allows one to follow the collapse of the foam, whichn the present case is caused by the superposition of drainage andoalescence.
In a previous FCA foam study with mixtures of non-ionic sur-actants it has been shown that an electrolyte concentration of ateast NaCl ∼10−2 mol l−1 is required to measure electrical conduc-
ivities of the respective foam and thus to measure the evolutionf the liquid fraction down to very low values [27]. On the otherand, foams stabilized by C12TAB (i.e. an electrolyte) had a highnough conductivity so that the liquid content of the respective
n be characterized by determining the average length of the Plateau border LPB bye length of the parietal Plateau borders LPPB (bold, red line), which, in turn, allows
2.6 mm prior to free drainage.
foams could be measured without adding NaCl. For the studies withthe mixtures the electrolyte concentration was adjusted such thatit corresponds to the electrolyte concentration of the pure C12TABsolution, which is 0.03 mol l−1 for 2 cmc and 0.15 mol l−1 for 10 cmc,respectively.
3. Results and discussion
Foams stabilized with a surfactant mixture consisting of thenon-ionic dodecyldimethyl phosphineoxide (C12DMPO) and thecationic dodecyl trimethylammonium bromide (C12TAB) with mix-ing ratios of C12DMPO:C12TAB = 1:0, 50:1, 1:1, 1:50, 0:1 have beenstudied with the home-built Foam Conductivity Apparatus (FCA).In the following the results obtained are separated into two mainsections, namely foams produced by the single surfactants (Section3.1) and by surfactant mixtures (Section 3.2). Section 3.1 examinesthe influence of the type of surfactant and of the total surfactantconcentration on drainage. Moreover, the influences of the bub-ble size and of the initial liquid fraction are examined. Section 3.2examines how the composition of the surfactant mixture at a fixedtotal concentration and how the bubble size and the initial liquidfraction of the 1:1 mixture influence drainage.
3.1. Free drainage of foams stabilized by single surfactantsC12DMPO and C12TAB
3.1.1. Influence of type of surfactant and total surfactantconcentration
In Fig. 2 the liquid fraction � is plotted as a function of timet for (top) C12TAB at c = 2, 5 and 10 cmc, and (bottom) C12DMPOat c = 2 and 10 cmc with a background electrolyte concentrationof 0.03 mol l−1 and 0.15 mol l−1. The concentrations were chosensuch that no depletion occurs during foam generation (see Sup-porting Material, Section S1) [38]. The drainage curves are fittedwith a power law ε∼ t� to learn something about the flow regimeas suggested in literature [2,14,15,20]. In all cases power law fittingover only one order of magnitude was possible as the liquid frac-tion quickly dropped to immeasurable small values. The exponent �provides information on the flow regime and it holds −2 < � < −1 formobile (plug-like flow) and −1 < � < −2/3 for immobile (Poiseuilleflow) surfaces [15]. For all systems studied in the paper at hand
values from −0.8 to −1.0 are found, which means that the surfaceshave low mobilities and that Poiseuille flow takes place. Note thatit is difficult to be in regimes with mobile surfaces. With standardfoams one usually finds immobile surfaces with exponents close
10 E. Carey, C. Stubenrauch / Colloids and Surfaces A
ε
t / s10 10 0 100 0 1000 0
10-4
10-3
10-2
10-1
100
C12
TAB
5 c mc, η = -0.8
10 cmc, η = -0.8
2 c mc, η = -0.8
- 2/3 (imm obile )
- 2 (mobile )
ε
t / s10 10 0 100 0 1000 0
10-4
10-3
10-2
10-1
100
C12
DMPO
10 c mc, η = - 0.8
2 c mc, η = - 1.0
- 2/3 (imm obile )
- 2 (mobile )
Fig. 2. Liquid fraction ε as function of time t for (top) C12TAB at c = 2, 5 and 10cmc and (bottom) C12DMPO at c = 2 and 10 cmc with a background electrolyte con-centration. Onset of drainage is at t = 0 s (not shown) with an initial liquid fractionεinit = 0.10–0.20 and an initial average Plateau border length LPB ∼ 1.0 mm. Mea-surements were recorded at foam height Hfoam = 20 cm. � is the exponent of thepower law fit of the ε(t)-curve. Sloped dash lines indicate the extremes of �, namely� = −2 for immobile and � = −2/3 for mobile surfaces. (top) Wall effects are observedfor C12TAB as indicated by a sharp change in the slope of the ε(t)-curve and, inturn, do not allow to determine a critical liquid fraction εcrit . (bottom) Critical liq-ud
tsbwmttnteb[w�a
without breaking [25]. Carrier et al. argued that not the pressure
id fractions εcrit were observed for C12DMPO and are indicated by the horizontalashed lines.
o −1. Indeed, Saint-Jalmes et al. found for pure sodium dodecylulfate (SDS) solutions exponents � from −0.8 to −1.3 for a bub-le diameter of 0.8 mm [14]. In this case the �-value increasedith foam height. Similar observations were made for measure-ents here within (see Supporting Material, Section S4). It is argued
hat the mobile surface regime could not be fully reached withhese bubble sizes but that larger bubbles are required. However,o information on the surfactant concentration used is given andherefore a direct comparison with our data is not possible. Carriert al. found for a 0.1 wt.% solution (∼115 cmc) of sodium dodecyl-enzenesulfonate (SDBS) at LPB = 1.2 mm an exponent of � = −1.32]. A value of � = −2 was only found for big bubbles (LPB = 2.3 mm),
hile for much smaller bubbles (LPB = 0.24 mm) the exponent was
= −0.8. The exponent � obviously depends on the foam heightnd on the bubble size. However studies on how � depends on the
: Physicochem. Eng. Aspects 419 (2013) 7– 14
surfactant concentration have not been published yet. To fill thisgap we carried out measurements at different concentrations. Theresults are presented in Fig. 2.
Note that in the case of C12TAB a ‘wall effect’ was observedfor c ≥ 2 cmc at t > 1100 s (see Supporting Material, Section S2)which is indicated by a sharp change in the slope of the �(t)-curvein Fig. 2 (top). Consequently, a critical liquid fraction εcrit, whichcorresponds to the liquid fraction at which foam rupture occurs,could not be determined. Interestingly, Carrier et al. studied freedrainage with a similar experimental set-up using tetradecyltrimethylammonium bromide (C14TAB) [2]. In this work a ‘walleffect’ is not mentioned not even at extremely high surfactantconcentration (c = 100 cmc). Whether or not the foam column wastreated to prevent such an effect cannot be extracted from thepaper. Monin et al. studied foams of dodecyl trimethylammoniumbromide (C12TAB) up to c = 6 × 10−3 mol l−1 = 0.4 cmc in the pres-ence of an anionic polyelectrolyte of various molecular weightsand degrees of charge [39]. They found that pure C12TAB in thisconcentration range produces very unstable foam. Alkyl trimethy-lammonium bromides adsorb strongly to negatively chargedsurfaces such as Plexiglas. The strong adsorption of C12TAB mayrender the surface of the Plexiglas column hydrophobic whichwould cause a dewetting and thus an additional foam destructionprocess. As expected, with increasing surfactant concentration thetime after which the ‘wall effect’ is first observed becomes shorter.Papara et al. have recently investigated the effects of the container(size, hydrophobic and hydrophilic nature) on free drainage of wetfoams [40]. Comparing a hydrophobic (untreated Plexiglas) and ahydrophilic (treated Plexiglas) container, one sees that foam decayis slower for the hydrophobic container – the difference gets lessevident if one increases the diameter of the container. Howeverthe lack of wetting analysis in both cases does not allow for adirect comparison. The fact that we observe the opposite trendis most likely due an excessive adsorption of C12TAB renderingthe columns wall extremely hydrophobic and thus dewetting thesurfaces immediately.
In the case of C12DMPO no ‘wall effect’ was observed. This, inturn, made it possible to determine the critical liquid fraction εcritin the presence of an electrolyte cNaCl ≥ 10−2 mol l−1. Collapse offoam resulted in an abrupt decrease in conductivity (and thus adramatic reduction in liquid fraction). Visual monitoring of the col-lapsing process confirmed that this conductivity jump was not asa result of being in a lower limit of conductivity measurably (seeSupporting Material, Section S3). Similar εcrit values are obtainedfor two different concentrations, namely εcrit = 0.0031 ± 0.0009 atc = 2 cmc (cNaCl = 0.03 mol l−1) and εcrit = 0.0034 ± 0.0005 at c = 10cmc. (cNaCl = 0.15 mol l−1). In this case the salt concentration wasvaried. Working at a concentration where the foam surface issaturated resulted in similar εcrit values. Indeed, Carrier et al.found for tetradecyl trimethylammonium bromide (C14TAB) thatεcrit decreases rapidly with increasing surfactant concentration atconcentrations around critical micellar concentration and almostsaturates at higher concentrations (c = 33–100 cmc) [1].
For polyhedral foams the disjoining pressure П is equal to thecapillary pressure with П =�/r = (0.6�)/(LPB
√ε) [1], where � is the
surface tension, r is the radius of curvature of the plateau borders,LPB is the average length of the plateau borders, and ε is the liq-uid fraction. Thus the capillary pressure П acting on the films inthe foam just before coalescence can be calculated as 320–390 Pa(using � = 29 mN m−1[25], LPB = 1.0 mm, εcrit = 0.003–0.002). Simi-lar to Carrier et al. [1] these values are orders of magnitude smallerthan the values that isolated horizontal films are able to support
but another mechanism must be responsible for coalescence indraining foams and suggested a mechanism based on critical filmdilatation [1].
E. Carey, C. Stubenrauch / Colloids and Surfaces A: Physicochem. Eng. Aspects 419 (2013) 7– 14 11
Table 1Initial liquid fraction εinit , critical liquid fraction εcrit , average Plateau border length LPB and NaCl content for samples of C12DMPO at c = 10 cmc. In all cases a total electrolyteconcentration of 0.15 mol l−1 was chosen. Measurements were performed at foam height Hfoam = 20 cm. Data extracted from Fig. 3.
C12DMPO:C12TAB = 1:0, 50:1 and 0:1 one finds that εcrit (i.e.the liquid fraction at which the foam ruptures) is lowest for the 1:1
TICaH
� −0.8 −1.0
.1.2. Influence of bubble size and initial liquid fractionThe effect of bubble size on free drainage of C12DMPO
t c = 10 cmc with a background electrolyte concentration ofNaCl = 0.15 mol l−1 was examined. In Fig. 3 (top) the averagelateau border length LPB is varied prior to drainage for the samenitial liquid fraction of εinit ∼ 0.08. As discussed in Section 2.2, apecific liquid fraction can be obtained by adjusting the imposediquid flow rate to the top of the foam column prior to measure-
ents. As expected the greater the bubble size of the producedoam the greater the imposed liquid flow rate required in ordero obtain the same εinit. The exponent � of the power law fits wasound to increase with increasing LPB (see Table 1). Increasinghe bubble size one also sees that the time at which the criticaliquid fraction εcrit is obtained decreases, i.e. the drainage times shorter. Thus foams consisting of small bubbles tend to drainlower compared to those with large bubbles. Looking at Fig. 3 oneees that it is hard to tell whether the bubble size has an effect onhe critical liquid fraction as the differences are very small. This isn line with literature [1], where it is found that the critical liquidraction εcrit does not depend on the length of the plateau bordernd that εcrit is a range rather than a specific value.
Fig. 3 (bottom) shows the influence of the initial liquid frac-ion εinit on free drainage of foams stabilized by C12DMPO at
= 10 cmc. Here foams with a constant average Plateau bor-er length LPB ∼ 1.2 mm were generated, while εinit was variedεinit = 0.16–0.03) by adding different amounts of the foaming solu-ion to the top of the foam column prior to drainage. The exponent
was found to decrease with decreasing εinit, which is in lineith investigations carried out by Saint-Jalmes et al. for foams of
odium dodecyl sulfate (SDS) with a much smaller bubble size ofPB ∼ 0.06 mm at εinit = 0.27–0.07 [14]. This observation is some-hat intuitive as foams of the same structure but higher initial
iquid fractions are expected to drain to a larger degree, which cane interpreted as foams with less rigid surfaces (larger �-values).
able 2nitial liquid fraction εinit , critical liquid fraction εcrit , average Plateau border length LPB
12DMPO:C12TAB = 1:0, 50:1, 1:1, 1:50 and 0:1 at a bulk surfactant concentration c = 2 cddition of NaCl such that it corresponds to 0.03 mol l−1 (∼=2 cmc C12TAB) and 0.15 mol lfoam = 20 cm. Data extracted from Fig. 4.
C12DMPO:C12TAB 1:0 50:1
2 cmcNaCl/mol l−1 0.030 0.030
LPB/mm 1.0 1.0
εinit 0.0926 0.1441
εcrit 0.0031 ± 0.0009 0.0018 ± 0.00� −1.0 −0.9
10 cmcNaCl/mol l−1 0.150 0.150
LPB/mm 1.0 1.0
εinit 0.1762 0.1600
εcrit 0.0034 ± 0.0005 0.0017 ± 0.00� −0.8 −0.8
−0.9 −0.8 −0.8
Note that the observed critical liquid fractions (see Table 1) are verysimilar and seem to be independent of εinit.
3.2. Free drainage of foams stabilized by mixtures of C12DMPOand C12TAB
3.2.1. Variation of composition at fixed concentrationFig. 4 shows ε(t)-curves for C12DMPO:C12TAB = 1:0, 50:1, 1:1,
1:50 and 0:1 at c = 2 cmc and 10 cmc for foams with the same ini-tial liquid fraction εinit ∼ 0.17 and average length of Plateau borderLPB ∼ 1.0 mm. A constant electrolyte concentration of 0.03 mol l−1
and 0.15 mol l−1 for 2 cmc and 10 cmc measurements, respectively,was adjusted. As the curves overlap to a large extent, the experi-mental y-data are multiplied by a factor for the sake of clarity (seeSupporting Material, Section S5 for unmodified curves). Selecteddata have been extracted and are presented in Table 2. The expo-nents � of the power law fits of the ε(t)-curves are similar thusindicating the same flow regime for all mixtures studied, namelyPoiseuille-like flow. The similar drainage behaviour observed forall mixing ratios is somewhat expected due to the similar structureof the surfactants (i.e. similar head group size and C12 hydropho-bic chain). This is especially true if the interactions between themixing components are weak, which was shown to be the case inprevious work [25]. Another reason for similar �-values and thusdrainage behaviour is the bulk viscosity �, which can be assumedto be the same in all cases. Indeed, Saint-Jalmes et al. demon-strated that � > 16 �water is required for a sodium dodecyl sulfate(SDS)/dodecanol (DOH) mixture before a change in the flow regimecan be observed [14].
Examining the critical liquid fraction εcrit of
mixture. Thus, as was the case for the FoamScan results, it appearsthat the 1:1 mixture has the highest stability [26]. Comparing
, exponent � of the power law fit of ε(t)-curve and NaCl content for samples ofmc and c = 10 cmc. In all cases the electrolyte concentration was adjusted via the
−1 (∼=10 cmc C12TAB), respectively. Measurements were performed at foam height
1:1 1:50 0:1
0.0294 0.016 –1.0 1.0 1.00.1579 0.1718 0.1580
06 0.0015 ± 0.0003 – –−0.8 −0.9 −0.8
0.147 0.078 –1.0 1.0 1.00.1860 0.1826 0.1826
09 0.0014 ± 0.0005 – –−0.8 −0.8 −0.8
12 E. Carey, C. Stubenrauch / Colloids and Surfaces A: Physicochem. Eng. Aspects 419 (2013) 7– 14
ε
C12
DMPO
c = 10 c mc
t / s10 10 0 100 0 1000 0
10-4
10-3
10-2
10-1
100
LPB
(vari ed)
εinit
(cons tant)
LPB
= 2.6 mm , η = -1.0
LPB
= 1.2 mm , η = -0.8
ε
t / s10 10 0 100 0 1000 0
10-4
10-3
10-2
10-1
100
C12
DMPO
c = 10 c mc
εinit
= 0.161 , η = -0.9
εinit
= 0.074, η = -0.8
εinit
= 0.027, η = -0.8
LPB
(cons tant)
εinit
(varied)
Fig. 3. Liquid fraction ε as function of time t for C12DMPO at c = 10 cmc. In all cases aconstant electrolyte concentration of 0.15 mol l−1 NaCl was used. Prior to the onset ofdrainage at t = 0 s a homogeneous foam is produced. Onset of drainage is at t = 0 s (notshown). Measurements were recorded at foam height Hfoam = 20 cm. (top) AveragePlateau border length LPB is varied at constant initial liquid fraction εinit of ∼0.08.(bottom) Average Plateau border length LPB is constant (LPB ∼ 1.2 mm) and initialliquid fraction ε is varied. Solid lines indicate the slope and thus the exponent �or
Ftic0CM
3m
td0biF
ε
t / s10 10 0 100 0 1000 0
10-4
10-3
10-2
10-1
100
101
102
103
104
1:0, η = -1.0
1:1, η = -0.8
1:50, η = -0.9
0:1, η = -0.8
50:1, η = -0.9
C12
DMPO:C12
TAB
c = 2 c mc x10
4
x102
x103
x101
x100
ε
t / s10 10 0 100 0 1000 0
10-4
10-3
10-2
10-1
100
101
102
103
104
1:0, η = -0.8
1:1, η = -0.8
1:50, η = -0.8
0:1, η = -0.8
50:1, η = -0.8
C12
DMPO:C12
TAB
c = 10 c mcx10
4
x103
x102
x100
x101
Fig. 4. Liquid fraction � as function of time t for C12DMPO:C12TAB = 1.0, 50:1, 1:1,1:50 and 0:1 at (top) c = 2 cmc and (bottom) c = 10 cmc. In all cases the elec-trolyte concentration was adjusted via the addition of NaCl such that c(electrolyte)= 0.03 mol l−1 and 0.15 mol l−1 for measurements at c = 2 cmc and c = 10 cmc, respec-tively. Prior to the onset of drainage at t = 0 s a homogeneous foam of constant liquidfraction is produced. Onset of drainage is at t = 0 s (not shown) with an initial liquidfraction εinit ∼ 0.10–0.17 and initial average Plateau border length LPB ∼ 1.0 mm.Measurements were recorded at foam height Hfoam = 20 cm. � is the exponent of thepower law fit of the ε(t)-curve for t ≤ 800 s. In the case of C12DMPO:C12TAB = 1:50
init
f a power law fit (� > −1 for immobile and � < −1 for mobile surfaces). Numericalesults are shown in Table 1.
oamScan with FCA results one always has to keep in mind that inhe former case all foams contained different liquid fractions, whilen the latter case measurements at same initial liquid fractionsould be carried out. In the case of C12DMPO:C12TAB = 1:50 and:1 a εcrit value was not found due to ‘wall effects’ as a result of12TAB adsorption along the foam column wall (see Supportingaterial, Section S2).
.2.2. Influence of bubble size and initial liquid fraction for 1:1ixture
In order to study the effect of bubble size on free drainage,he average length of the Plateau border LPB is varied prior torainage for C12DMPO:C12TAB = 1:1 at c = 10 cmc in the presence of
−1
.147 mol l NaCl. The results are shown in Fig. 5. Here it shoulde noted that an increase in the bubble size LPB leads to a decrease
n the initial liquid fraction εinit (thus both LPB and εinit are varied).uture work involves the development of online measurements
and 0:1 wall effects are observed and are indicated by a sharp change in the slopeof the ε(t)-curve. Numerical results are shown in Table 2. The experimental y-dataare multiplied by a factor for the sake of clarity.
of the initial liquid fractions during experimental runs, which, inturn, will allow for studying the influence of the bubble size LPBat constant initial liquid fraction. Generally it is argued that largerbubbles are more fragile than smaller bubbles [41–44] and that thestability is inversely proportional to the surface area of the bubble[45]. However Vandewalle et al. showed through acoustic exper-iments that small and large bubbles are involved in the avalancheprocess during the dynamics of a collapsing foam and suggestedthat the radii of the bubbles do not govern the stability of thedraining foam [46,47]. From Figs. 3 and 5 it can be concluded thatdrainage is slower when the bubbles of the foam are smaller. This
is in line with previous free drainage experiments [1,20,48–50].The jump of liquid fraction at the end of the experiments, i.e. at thearrival of the rupture front, is more significant for big bubbles thanfor small bubbles [1,2]. Therefore the evolution of liquid fraction
E. Carey, C. Stubenrauch / Colloids and Surfaces A
t / s1 10 10 0 100 0 1000 0
10-4
10-3
10-2
10-1
100
εC
12DMPO: C
12TAB = 1:1
c = 10 c mc
LPB
= 0.3 mm , η = -0.8
LPB
= 2.3 mm , η = -1.0
LPB
= 1.0 mm , η = -0.8
Fig. 5. Liquid fraction ε as function of time t for C12DMPO:C12TAB = 1:1 at c = 10 cmc.In all cases a constant electrolyte concentration of 0.15 mol l−1 NaCl was used. Priorto the onset of drainage at t = 0 s a homogeneous foam of constant liquid fraction isproduced. The average Plateau border length LPB and the initial liquid fraction εinit
are varied. Measurements were recorded at foam height Hfoam = 20 cm. Solid linesindicate the slope and thus the exponent � of a power law fit (� > −1 for immobileand � < −1 for mobile surfaces). Numerical results are shown in Table 3.
Table 3Initial liquid fraction εinit , critical liquid fraction εcrit , average Plateau border lengthLPB and NaCl content for samples of C12DMPO:C12TAB = 1:1 at a bulk surfactantconcentration c = 10 cmc. In all cases the total electrolyte concentration of 0.15 moll−1 was adjusted by the addition of NaCl. Measurements were performed at foamheight Hfoam = 20 cm. Data extracted from Fig. 5.
s less continuous in the case of large bubbles than in the casef smaller bubbles. This was visually observed. In the case of bigubbles, rupture occurred in large avalanches whereas the heightf foams consisting of small bubbles decreased in small steps.imilar to the findings of Carrier et al. [1] the threshold in liquidraction was found to be independent of the size of the bubbles (seeig. 5 and Table 3) and the foam height Hfoam (see Section 3.1.1),ith destruction occurring in a narrow range of liquid fraction.
The exponent � of the power law fits of the ε(t)-curve was foundo increase with increasing LPB (see Table 3), which is in line withrevious studies [2]. Thus an increase of the bubble size rendershe surface more mobile if all other parameters are kept constant.nterestingly, it was found that for C12DMPO:C12TAB = 1:1 at c = 10mc the surfaces remained immobile (vanishing flow velocity in thealls and Poiseuille flow) from LPB = 0.3 mm up to 2.3 mm. Saint-
almes et al. showed that for a moderate surface viscosity (SDS/DOHolution), one can have a change from plug flow to Poiseuille flowegimes, and back again to plug flow, as one decreases the bubbleize [14]. However as already mentioned, it is usually difficult to ben regimes with mobile interfaces, i.e. in the plug flow regime.
The free drainage curve obtained for foam with an averagelateau border length LPB = 0.3 mm (i.e. small bubbles) appears toave three different regions (see Fig. 5). Firstly, the liquid fraction εemains almost constant with time t (t ≤ 100 s) for a certain holdup
: Physicochem. Eng. Aspects 419 (2013) 7– 14 13
time th as a result of capillary suction caused by differences incurvature along the bubble. This delayed foam drainage dependson the bubble size and the initial liquid fraction [51,52]. Indeedfor measurements with different initial liquid fractions εinit (Fig. 3(bottom)) the delayed drained period is greatest for the lowestεinit = 0.027 (t ≤ 80 s). This can be attributed to smaller fluid veloc-ity [52]. For the second region a typical steady decrease of ε witht (100 ≤ t ≤ 7000 s) with a distinctive slope is found indicatingfree drainage. Finally a notable change in �(t) curve prior to rup-ture (t ≥ 7000 s) is observed. Similar three-step curves during freedrainage are observed elsewhere for bubbles of similar size (LPB <0.6 mm) [1,2], however with a distinctive upturn in the slope priorto collapse. In Fig. 5, the distinctive downturn at t ≥ 7000 s suggestsstrong coalescence of this quite weak foam which, in turn, leads toa quick collapse.
4. Conclusions and outlook
We used the Foam Conductivity Apparatus (FCA) to examinefoams stabilized by C12DMPO, C12TAB, and their mixtures. Theexponents � of the power law fits of the ε(t)-curvesare similar(h = −0.8 to −1.0) for all samples studied, which indicates thesame flow regime, namely Poiseuille-like flow. The similar drainagebehaviour can be explained qualitatively with similar surfactantstructures and bulk viscosities. However, it was not expected thatthese surfactants and their mixtures have immobile surfaces underall experimental conditions. Note that for a quantitative analysisthe surface viscosities and surface elasticities are needed. In a pre-vious study with the two non-ionic surfactants C12E6, �-C12G2 andtheir 1:1 mixture [53,54] we found that a higher rigidity (extractedfrom the drainage curves) goes in parallel with a higher elastic-ity (measured with an oscillating bubble tensiometer). In order tostudy the surface rheological behaviour for the present system oneneeds to carry out a systematic oscillating bubble study combinedwith capillary wave measurements since the concentrations of thecationic surfactant is too high for oscillating bubble measurements.This, however, was beyond the scope of the study at hand.
Although for all cases similar � values were found, the foam sta-bilities are different, i.e. drainage is not the factor that determinesthe stability. In line with literature[1] a critical liquid fraction εcritrange was found, which is independent of foam height and bubblesize. In the case of C12DMPO:C12TAB = 1:50 and 0:1 a εcrit valuewas not found due to ‘wall effects’ as a result of C12TAB adsorp-tion along the wall of the foam column. For the studied mixturesdrainage leads in all cases to dry foams (ε ∼ 10−3), which theneither coalesce (C12DMPO and 50:1 mixture) or drain further (1:1mixture). The stability of foams generated by the 1:1 mixture wasgreater than that of C12DMPO and of the 50:1 mixture, which canbe explained by the fact that at this specific mixing ratio NBFs areobserved at all studied bulk concentrations [25].
In addition, the effects of the average Plateau border length LPBand the initial liquid fraction εinit on the drainage curves has beenstudied. An increasing LPB leads to shorter drainage times and to anincrease of the exponent � of the power law fits. On the other hand,the critical liquid fraction εcrit is not affected. An increase of εinitleads to longer drainage times and to an increase of the exponent� of the power law fits, while again the critical liquid fraction εcritis not affected. In general, drainage times are affected to a largerextent by LPB compared to εinit.
In conclusion, our systematic study of well-defined foams sta-bilized by surfactant mixtures showed similar drainage behaviourbut a significant difference in the overall foam stability. Following
drainage to very low liquid fractions, we found that the coales-cence of foams stabilized by the 1:1 mixture occurred at muchslower rates. We assigned this stability to the formation of NBFas reported in our previous work [25]. Thus the mixing ratio of a
1 aces A
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4 E. Carey, C. Stubenrauch / Colloids and Surf
urfactant mixture, even in the absence of synergistic effects, canave a dramatic influence on the overall foam stability withoutecessarily affecting the drainage behaviour.
cknowledgments
E.C. would like to acknowledge UCD Ad Astra Research Schol-rship funding. Part of the work was funded by the Europeanommunity’s Marie Curie Research Training Network ‘‘Self-rganisation under Confinement (SOCON)”, contract numberRTN-CT-2004-512331. We thank Dr. Wiebke Drenckhan for very
elpful discussions.
ppendix A. Supplementary data
Supplementary data associated with this article can beound, in the online version, at http://dx.doi.org/10.1016/j.colsurfa.012.11.037.
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