Dissolution Dynamics of Liquid/Liquid Binary Mixtures

Transp Porous MedDOI 10.1007/s11242-013-0223-1

Dissolution Dynamics of Liquid/Liquid Binary MixturesWithin a Micromodel

Mihaela S. P. Stevar · Anatoliy Vorobev

Received: 8 March 2013 / Accepted: 28 August 2013© Springer Science+Business Media Dordrecht 2013

Abstract We report optical observations of the dissolution behaviour of glycerol/water, soy-bean oil/hexane, and isobutyric acid (IBA)/water binary mixtures within a glass micromodelbuilt as a 2D regular network of capillary tubes with diameter of 0.2 mm. The micromodel isinitially filled with solute and then is horizontally immersed into a thermostatic solvent-filledbath. The micromodel is open at its corners for solute dissolution to occur with no pressuregradients being applied. Our study shows that the solvent penetration into the micromodelis diffusion-dominated in completely miscible binary mixtures (glycerol/water and soybeanoil/hexane). This is, however, non-Fickian diffusion with the dissolution rate, dV /dt, beingproportional to D1/3t−0.4 for almost the entire duration of the experiment (V is the volumeoccupied by the solvent, D is the diffusion coefficient, and t is time). For the partially mis-cible IBA/water mixture the experiments performed at undercritical temperatures revealedthat the diffusive transport was negligible despite the mixture being out of its thermodynamicequilibrium. The water phase penetrated into some of the channels, but IBA was never com-pletely displaced/dissolved from the micromodel and numerous interfaces remained visibleafter very long-time periods.

Keywords Binary mixture · Liquid/liquid phase transition · Dissolution rate ·Non-Fickian diffusion · Micromodel

1 Introduction

Over the past few decades, multiphase flows in porous media have been extensively studiedthrough both experiments and numerical simulations due to their large-scale applications(Sahimi 1993; Vanaparthy and Meiburg 2008) in enhanced oil recovery (EOR) (Donald-son et al. 1985; Babadagli 2007), enhanced aquifer (and soil) remediation (EAR) (Jawitz

M. S. P. Stevar · A. Vorobev (B)Energy Technology Research Group, Faculty of Engineering and the Environment,University of Southampton, Southampton SO17 1BJ, UKe-mail: [email protected]

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M. S. P. Stevar, A. Vorobev

et al. 1998; Silva et al. 2005), solvent extraction (Mukhopadhyay 2000) and drug delivery(Dokoumetzidis and Macheras 2006).

Since the pore sizes are typically small, this makes the hydrodynamic flows within porousmedia being rather slow and as a result the diffusion, wetting and capillary effects playessential roles in the overall mass transport. The porous medium can be roughly representedas a network of interconnected capillary tubes (Aker et al. 1998; Blunt et al. 2002; Lovoll et al.2005). As pore-scale network modelling on its own cannot provide all the detailed informationneeded for practical problems to be solved at macroscale, the necessity of completing thesestudies brought forward the experimental use of micromodels for visualizing flow patternsat the pore level.

Micromodels are 2D transparent networks of pores and constrictions widely used forthe study of flow processes at the pore level. Micromodels can simulate the complexity ofnatural porous media just up to a certain degree (Jeong et al. 2000), but they allow tracking theinterfaces position and movement through the porous network, which makes them valuableutensils for multiphase flow studies.

Numerous experimental studies which employ micromodels for studying a large varietyof pore-scale mechanisms characteristic to EOR and EAR processes have been reported inthe literature in the last few decades (Jia et al. 1999; Sahloul et al. 2002; Sohrabi et al. 2008;Armstrong and Wildenschild 2012). Besides petroleum engineering and geological applica-tions there are many other fields in which micromodels find their applicability, as for instancedrying technology (Oyarzun and Segura 2009). The use of micromodels increased continu-ously after 1980 and exploded after 2000 when better techniques became available for theirmanufacturing. More details about the successes and the limitations of micromodels in illus-trating the mechanisms of multiphase ow, the approaches employed to design and fabricatemicromodels, fabrication techniques, applications and examples of various micromodels canbe found in the reviews of Buckley (1991) and Blunt (2001). The most commonly employedtechnique to fabricate micromodels is the etching. Glass micromodels are considered to bemore advantageous from many points of view including optical (Jamaloei et al. 2010a) andmechanical properties, chemical and thermal resistance, surface coatings, design flexibility,durability, reusability and cost effectiveness. The large number of publications with resultsof microvisual experiments shows that despite their limitations micromodels are very usefuland continue to find new applications. Still, their contribution is mostly qualitative rather thanquantitative. The disadvantage of larger models is that they are more difficult to manufacturebut on the other hand they might have the advantage of reducing boundary effects and theycould provide better statistical samples for comparison with simulations data. Still, it remainsa challenge to create more realistic models of the porous media while retaining visibility.

Using the so-called network approach (Fig. 1), we define the aim of the current work as tounderstand the dissolution dynamics of miscible binary mixtures within a regular flow-pathmicromodel, which represents a step further from our previous work (Stevar and Vorobev2012), in which the dissolution behaviour of liquid/liquid miscible interfaces was investigatedin single capillary tubes.

In this work, the micromodel saturated with the solute phase was immersed into a solvent-filled thermostatic bath. When the pure components of a binary mixture are brought intocontact a non-equilibrium binary system is obtained. Equilibration of the mixture to a thermo-dynamically stable state involves diffusive and hydrodynamic processes, and usually occursvery slowly. The glass micromodel used in this work has dimensions comparable with thetypical pore sizes in geological applications frequently assumed to be 0.1 mm and smaller.However, the pores can be larger in the case of vegetable oil extraction, and hence the resultspresented are even more relevant to such applications.

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Dissolution Dynamics of Liquid/Liquid Binary Mixtures

Fig. 1 Modelling of porous medium by a network of pores and throats

The focus of this study is on the dissolution process, namely, the mass exchange betweenthe components of binary mixtures, which is traditionally modelled as a process of inter-diffusion. A heterogeneous binary mixture is in equilibrium if the components have equalchemical potentials, when the interfacial mass transfer is absent despite the strong differencein concentrations. In the absence of temperature gradients, the rate of dissolution in a non-equilibrium system is determined by the difference in the chemical potentials between theneighbouring phases. Since the chemical potential of an isothermal system is in general afunction of concentration and pressure, then, in addition to the Fickian diffusion determinedby concentration gradients, the mass transport can also be driven by pressure gradients (Lan-dau and Lifshitz 2009). Variations in pressure field can be induced by gravity and capillaryforces and thus can introduce a new diffusion mechanism called barodiffusion (Landau andLifshitz 2009; Viner and Pojman 2008; Vorobev 2010). Despite an apparent simplicity ofthe diffusion it is difficult to provide a simple estimate for the typical dissolution time of asolute droplet surrounded by solvent (Stevar and Vorobev 2012), and the problem becomeseven more challenging if a liquid/liquid solute/solvent system saturates a porous bulk. Inthe past few decades, there have been a large number of theoretical and experimental stud-ies on diffusion in porous media but most of them were concerned with gaseous diffusion(Currie 1960a,b, 1961; Zheng et al. 2012) or with diffusion of weak impurities (Delgado2007). In most such dispersion studies, diffusion coefficients and velocities were measuredby employing NMR techniques (Merrill and Jin 1994; Rajanayagam et al. 1995).

In gaseous systems, molecules are disperse and as a result, intermolecular interactions areweak. Diffusion occurs due to the random walk of molecules. The scenario becomes differentin liquid mixtures due to the involvement of intermolecular forces, which are different withineach phase. The difference in these forces forms a potential barrier which molecules need tocross for interfacial mass transfer to occur. In the case of immiscible liquids, this barrier is sohigh that no molecules are able to overcome it, while in the case of miscible systems somemolecules are able to cross the potential barrier, but its existence results in a lower interfacialmass flux compared with the free diffusive flux.

Diffusion becomes even more complicated if the liquid mixture saturates a porous medium,as in this case both, the geometry of the pores and the interactions with the solid wallscan affect molecules trajectories. To this end, the term anomalous diffusion (also referredto as restricted diffusion) is introduced (Sen 2003; Kim et al. 2007; Fomin et al. 2010,2012). Anomalous diffusion is usually further divided into subdiffusive (slower than Fickiandiffusion, σ < 1) and superdiffusive (faster than Fickian diffusion, σ > 1) regimes. In thesecases the root-mean-square value of the displacement of the particles is not proportional withtime,

(�x)2 ∝ tσ , (1)

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as in the case of Fickian diffusion (σ = 1) (Jou et al. 2011). Both regimes of anom-alous diffusion can be identified in natural porous media due to their very complexmicrostructures.

In addition, liquid/liquid interfaces are endowed with capillary properties, while liq-uid/solid interfaces should posses wetting properties (Joseph and Renardy 1993; May andMaher 1991; Pojman et al. 2006; Stevar and Vorobev 2012), which can introduce addi-tional mass transfer mechanisms. In our earlier experiments with single capillary tubeswe found that the mass transfer is predominately diffusion defined in liquids misciblein any proportion, and is driven by the capillary pressure in IBA/water binary mixturebelow its critical temperature. In both spontaneous imbibition and drainage the drivingforce of the displacement is the capillary pressure with the difference that in the formercase a (more) wetting fluid displaces a non-wetting one and in the latter case a non-wetting phase displaces a wetting phase. Even though a high number of studies involv-ing capillary-driven displacements within micromodels can be found in the literature, todate, the only other studies known to the authors that describe spontaneous (free) imbi-bition/drainage visual observations within micromodels are the ones of Chatzis and Dul-lien (1983) and Jamaloei et al. (2010a,b). We are unaware of any experimental work inwhich the dissolution dynamics of miscible binary systems saturating micromodels has beenstudied.

2 Materials and Methods

2.1 Binary Mixtures

In this work, we examine the evolutions of glycerol/water, soybean oil/hexane and isobutyricacid (IBA)/water binary mixtures within a micromodel. All chemicals having high puritieswere purchased from Fisher Scientific or Sigma Aldrich and were used as received.

Glycerol and water and soybean oil and hexane are miscible in any proportions at alltemperatures. The IBA/water mixture is characterized by the upper critical solution tem-perature, T = 26.2 ◦C (Chu et al. 1968; Venkataraman and Narducci 1977; Pojman et al.2006). At temperatures above the critical point, the mixture components are miscible in anyproportions. Below the critical point, only small amounts of IBA are completely miscible inwater, and if a larger droplet of IBA is added, then IBA and water phases will inter-diffuseuntil the equilibrium concentrations, c1 and c2, are established. The values of c1 and c2 aretemperature dependent and can be obtained from the phase diagram available, e.g., in thepaper of Pojman et al. (2006).

Values of diffusion coefficient for the glycerol/water mixture at different temperaturesand for different compositions can be found in the papers of Petitjeans and Maxworthy(1996); Rashidnia and Balasubramaniam (2004) and D’Errico et al. (2004). Just for ref-erence, the mutual diffusion coefficient at room temperature equals 1.6 × 10−10m2 s−1 atan interface between pure glycerol and water (Petitjeans and Maxworthy 1996). For soy-bean oil/hexane mixture, only theoretical estimations of the mutual diffusion coefficient aregiven by Wu and Lee (1999), being 9.1 × 10−10m2 s−1 at 25 ◦C. The IBA/water mutualdiffusion coefficient is zero in the critical point. Above the critical point the diffusion coef-ficient grows and equals about 1.3 × 10−11m2 s−1 at 27 ◦C (Kaatze and Mirzaev 2000).The values of density and viscosity coefficient for the pure substances are summarized inTable 1.

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Table 1 Density and viscositycoefficients for pure glycerol,soybean oil, IBA, water andn-hexane at 20 ◦C (Stevar andVorobev 2012)

Substance ρ (g cm−3) η (Pa s)

Glycerol 1.26 1.41

Soybean oil 0.917 0.069

IBA 0.95 1.1 × 10−3

Water 1.00 1.01 × 10−3

n-Hexane 0.659 0.326 × 10−3

Fig. 2 Micromodel used inexperiments with dimensions of45 × 45 × 4 mm, containing 19× 19 channels with near-circularprofile (0.21 mm width and0.2 mm depth) spaced at 2 mmdistance and a larger inletchannel on the edge (0.5 mmwidth and 0.2 mm depth); aempty micromodel; b sectionview of the channels (as suppliedby the manufacturer)

2.2 Micromodel

The regular pore-shape glass micromodel used for the experiments described in the currentwork was manufactured by the Dolomite Centre Ltd., UK using the isotropic wet-etchingtechnique. As the isotropic etching results in widening of features exposed from a mask theshape of the resulting channels corresponds to a near circular profile (with width 210µm anddepth 200µm). We considered this to be acceptable for our purpose as in the experimentsperformed within capillary tubes we obtained similar results for circular and square cross-sections (Stevar and Vorobev 2012). The length of the mutually perpendicular channelsconnecting the pores is 2 mm. The coordination number of the pores is equal to 4. Thedimension of the micromodel is 45 × 45 × 4 mm and it contains 361 pores and a largerchannel (500µm) on the edge to allow fluid to enter the porous media at any point (seeFig. 2). All micromodel characteristics are summarized in Table 2. The optical glass B270used for the micromodel fabrication offers excellent optical properties and chemical resistanceallowing a broad range of solvents and chemicals to be used.

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Table 2 Micromodelcharacteristics

Characteristic Value

Length 4.5 cm

Width 4.5 cm

Etched depth 200 µm

Pore and channel width 210 µm

Coordination number 4

Aspect ratio 1

Areal porosity 17 %

2.3 Experimental Rig

First, the micromodel was saturated with the solute phase (glycerol, soybean oil or IBA)and then immersed into a thermostatic transparent solvent-filled (water or hexane) bath.The micromodel was placed horizontally in the bath. No pressure gradient was applied,with all 4 corners of the micromodel being open for the solvent to penetrate into themicromodel.

The experiments were performed at various temperatures in the range of 20–30 ◦C. Thetemperature in the water bath was controlled with a Grant digital thermostat (GD100, stabilityat 37 ◦C is 0.05 ◦C); the homogeneity of the solvent temperature was checked in severalreference points with a Checktemp1 thermometer by Hanna Instruments (accuracy 0.3 ◦C).

The dissolution front was recorded with a system including a CCD camera (LaVi-sion Imager 3S) equipped with a Nikon AF Micro-Nikkor lens 60 mm f/2.8D, a dif-fuser (LaVision, VZ illumination high-efficiency diffuser) and a diode laser (Oxford-laser Firefly system) used for illumination. To improve the image contrast the solute-liquids were coloured with methylene blue, Sudan IV, or eriochrome black T dyes pur-chased from Sigma Aldrich. For visualizing the shapes of the interfaces in the pores andthroats of the micromodel, the Nikkor lens was replaced with a Questar lens (QM100Model #30003, with working distances in the range 15–35 cm, and magnification up to34 times at the image plane. A schematic view of the experimental setup is depicted inFig. 3.

3 Results and Discussion

3.1 Glycerol/Water Mixture

Glycerol and water are completely miscible in all proportions in any conditions. We observedthe propagation of glycerol/water diffusion fronts within a network model with a regularflow-path pattern. Some air bubbles remained trapped in the micromodel, which seeminglyhappened due to the filling procedure. The number of air bubbles was, however, rathersmall and could be disregarded as it did not affect any integral characteristics such as volumedistribution in the entire micromodel. Still, their presence was important at pore-level leadingto the non-symmetry of the glycerol/water boundary.

Several snapshots showing the progression of the glycerol/water boundary are depictedin Figs. 4 and 5 for the experiments carried out at 20 and 30 ◦C. In these figures, frames4g–l and 5g–l, respectively, show the interpretations of the experimental observations. The

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Fig. 3 A schematic view of the experimental setup

contours shown in these frames are based on data points obtained by averaging the amountof glycerol remained in a unit cell of the micromodel at the time moment indicated on eachframe. A unit cell of the micromodel is represented by a square with side channels of 2mm length. The amount of glycerol remained in each unit cell was calculated as Vg =(V1 + V2 + V3 + V4)/4 where Vi is the volume fraction of glycerol in every side channel(i = [1, 4]). Vi varies from 1 (i.e., fully filled with glycerol) to 0 (i.e., fully filled with water)and takes intermediate values in the range of 1–0 for the case of partially filled channels.For example, a schematic representation of a unit cell filled with glycerol, half glycerol/halfwater and water, respectively, is shown in Fig. 6.

The solvent penetrated uniformly into the micromodel indicating that the solute is dis-solved rather than displaced. The mass transport is diffusion dominated. As noticed, eventhough the micromodel employed in the experiments is a homogeneous network the expectedsymmetry of the diffusion front was not observed owing to the trapped air bubbles. The glyc-erol/water boundary advanced past these bubbles signifying that water could bypass thebubbles but could not displace them.

Figure 7 shows the total volume of water-rich phase within the micromodel at differenttime moments (defined as V = 1 − Vg). As expected the temperature increase shortens thedissolution time (Fig. 7a). Complete dissolution of glycerol from the micromodel took onaverage 42 h at 20 ◦C and around 30 h at 30 ◦C with complete dissolution times varying slightlybetween different experiments due to the amount and position of air bubbles trapped in themicromodel. However, as it can be seen in Fig. 7a the distribution of complete dissolutiontimes is quite narrow and therefore considered acceptable. In Fig. 7b it can be seen thattwo power laws, t0.3 and t0.6, can be identified to characterize the dissolution rate, butnone of them corresponds to Fickian law time dependence. The second regime is observedafter approximately 7000 s. The coefficients of proportionality, A and B, introduced by therelations V = At0.3 and V = Bt0.6, are 0.13 m3 s−0.3 and 8×10−4 m3 s−0.6 at 20 ◦C and0.14 m3 s−0.3 and 9.8×10−4 m3 s−0.6 at 30 ◦C.1

1 Note that the conclusion about the value of the first exponent is based on a fewer number of data points. Forthe second exponent, the detailed analysis gives the value 0.582 ± 0.016.

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Fig. 4 a–f Snapshots of the glycerol/water phase boundaries within micromodel; the channels filled withglycerol phase are shown with a lighter grey colour and the ones filled with water have a darker grey colour;air bubbles trapped within the channels are shown in black. g–l Interpretations of the experimental observations;the legend shows the volume fraction of glycerol in the mixture; T = 20 ◦C. The time moments are indicatedin the pictures

In our previous work, Stevar and Vorobev (2012) observed that the position of the glyc-erol/water front was moving into the capillary tube with the dependence t2/3 in the beginningof the experiment, when water entered the capillary tube and the convective motion could benoticed, and with t1/3 dependence characterizing the main diffusive run. When the speedsof the phase boundaries were considered these time dependencies became t−1/3 and t−2/3.The former expression characterized the initial evolution which was gravity-driven and thelatter one characterized the evolution possibly driven by barodiffusion.

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Fig. 5 a–f Snapshots of the glycerol/water phase boundaries within micromodel; the channels filled withglycerol phase are shown with a lighter grey colour and the ones filled with water have a darker grey colour;air bubbles trapped within the channels are shown in black. g–l Interpretations of the experimental observations;the legend shows the volume fraction of glycerol in the mixture; T= 30 ◦C. The time moments are indicatedin the pictures

The data collected in the current experiments also show that the time evolution can bedivided into two stages, with an initial transient trend being t0.3, and with a later main trendbeing t0.6. If the speed of the diffusion front is considered these dependencies become t−0.7

and t−0.4. It can be noticed that the former expression obtained in this case is very close tothe one obtained for the dissolution stage possibly driven by barodiffusion in single capillarytubes. The small difference in index value may be attributed to the different topology. Thelatter expression, t−0.4 (also different from the classical diffusion t1/2 dependence) charac-

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Fig. 6 Schematic representation of a unit cell of the micromodel; a volume fraction of glycerol in the cell is1; b volume fraction of glycerol remained in the cell is 0.5; c volume fraction of glycerol remained in the cellis 0

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terizes the second stage of dissolution within micromodel which represents a subdiffusiveregime. This second stage is much longer compared to the initial transient stage. Both regimesidentified in these experiments are characterized by power laws that differ from the classicalFickian dependence which signifies that Fick’s law is not capable of defining the observedbehaviour, not even in a porous medium with a high degree of homogeneity, and not evenwhen the dependence of the mutual diffusivity on concentration is taken into account.2

In order to take a closer view at the shape of the interface the experimental setup wasmodified as described in Sect. 2.3. For glycerol/water mixture the shape of the interfacecould be clearly distinguished for up to 10 min from the start of the experiment (Fig. 8a,b) and after that the tip could not be clearly observed due to its diffusive smearing, but thedissolution front could still be followed (Fig. 8c, d) up until the end of the experiment asalready shown in Figs. 4 and 5.

2 The mutual diffusion coefficient in glycerol-rich phase is known to be about 10-fold lower than in water-richphase (Petitjeans and Maxworthy 1996; D’Errico et al. 2004). Such a strong dependence allows the front-typepenetration of a solvent into a solute-saturated tube to be mathematically reproduced on the basis of the Fickianlaw, but the resultant front speed would be always proportional to t0.5 if only Fickian transport is taken intoaccount (Crank 1976). The front-type dissolution in polymer mixtures with the front speeds different fromt0.5 is known, see e.g., Cohen and White (1991), where such a behaviour is associated with viscoelasticproperties of polymers, but such an explanation and a developed mathematical model is not acceptable for theglycerol/water mixture.

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Fig. 8 Magnified view within the micromodel during an experiment with glycerol/water at different timemoments; a, b top view of the interface at early time moments; c the tip of the interface is already verydiffused; the arrows indicate its approximate position which can be determined based on the colour contrast;d the two phases can be identified based on the colour contrast; the channel filled only with water phase has adarker grey colour, the ones filled only with glycerol phase have a lighter grey colour and the channels whereboth phases are present are represented with lighter grey colour in the middle and darker grey colour on thechannel edges. In all frames ‘G’ denotes glycerol phase and ‘W’ denotes water phase. Temperature is T =20.5 ◦C. The corresponding time moments are indicated in the pictures

3.2 Soybean Oil/Hexane Mixture

Soybean oil and hexane are completely miscible in all proportions under any conditionssimilar to glycerol and water. We performed the same experiments as for glycerol/waterat both 20 and 30 ◦C. Similar data were obtained with just a few particularities. More airbubbles remained trapped in the micromodel during the solute-filling phase. This happenedmost likely due to the different wetting properties of soybean oil relative to glass comparedto the ones of glycerol. Moreover during the dissolution process these bubbles moved andformed larger aggregates that remained trapped in the network at the end of the experiment(Figs. 9, 10). The dissolution times for soybean oil/hexane were shorter compared to theones for glycerol/water (on average the dissolution took 19 h at 20 ◦C and 11 h at 30◦C,respectively) due to the higher diffusion coefficient of this mixture (Fig. 11a). Similarly theasymmetry of the diffusion front was observed. It is also worth mentioning that even thoughthe amount of air present in the micromodel was much higher (i.e., approximately doublecompared to the glycerol/water experiments) the diffusion front still continued its movementuntil the whole micromodel was filled with the solvent-phase.

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The dissolution process could be characterized with the same two power laws as in thecase of glycerol/water (see Figs. 7b, 11b). The coefficients, A and B, introduced by therelations V = At0.3 and V = Bt0.6, are 0.019 m3 s−0.3 and 1.3×10−3 m3 s−0.6 at 20 ◦Cand 0.02 m3 s−0.3 and 1.6×10−3 m3 s−0.6 at 30 ◦C.3 By comparing the B values for the

3 Note that similarly to the glycerol/water experiment, we may firmly state that there are two trends, and thatthe value of the first exponent is close to 0.3; for the second exponent, the detailed analysis gives the value0.591 ± 0.035.

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glycerol/water and soybean oil/hexane experiments one finds that the diffusion front in thesoybean oil/hexane experiments moves 1.6–1.7 times faster which agrees with the similardata obtained in the experiments with single capillary tubes (Stevar and Vorobev 2012). Bytaking into account the ratio of diffusion coefficients for these two binary mixtures, we canconclude that the dissolution rate is proportional to D1/3.

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0.7

0.8

0.9

1(a)

t (s)V

103 1040.1

0.2

0.3

0.4

0.5

0.6

0.70.80.9

1

t0.3

t0.6

(b)

Fig. 11 a The volume fraction of hexane within the micromodel at different time moments; b the same dataset plotted in logarithmic coordinates; filled circle T = 20 ◦C and filled square T = 30 ◦C

Fig. 12 Magnified view within the micromodel during an experiment with soybean oil/hexane showing thesame intersection of the channels at different time moments; a all channels are filled only with soybean oiland are represented by a lighter grey colour; ‘S’ denotes soybean oil phase; b all channels are filled only withhexane and are represented by a darker grey colour; ‘H’ denotes hexane phase. T = 30 ◦C

For soybean oil/hexane mixture the shape of the interface could not be clearly distin-guished, not even in the early time moments of the experiments. At early time moments thetip of the interface was already very diffused and could not be clearly identified. Still, similarto the case of glycerol/water the diffusion front could be easily followed based on the colourcontrast (Fig. 12) up until the end of the experiment (i.e., when the whole micromodel wasfilled with hexane) as already shown in Figs. 9 and 10.

3.3 Isobutyric Acid/Water Mixture

IBA/water is a partially miscible binary mixture with an upper critical solution tempera-ture at ∼26.2◦C. As already described in our previous work (Stevar and Vorobev 2012), thebehaviour of this mixture below and above the critical point is completely different. Abovethe critical point the interfaces become very diffused in a very short period of time with the

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Fig. 13 Isobutyric acid/water interphase boundaries within micromodel at different time moments; T = 22 ◦C

water phase underriding the IBA phase and thus the current method could not be used forinvestigation of the dissolution above the critical point. Below the critical point, in a singlecapillary tube the displacement of the IBA phase was governed by the capillary force (spon-taneous imbibition). Within the micromodel several interfaces were observed for IBA/watermixture but the contrast of the images obtained was not good enough to describe accuratelythe dissolution process. Therefore we decided to take a closer view and changed the Nikkorlens with the Questar one. The shapes of the interfaces in pores and throats are depicted inFig. 13. As it can be seen in this figure, numerous interfaces were observed which movedin different parts of the micromodel, but no clear pattern could be deduced. Interfaces werestill present within the micromodel even after long periods of time from the beginning of theexperiment (> 24 h, Fig. 13f–h).

Surprisingly, in contrast to our previous observations within single capillary tubes, thecurvature of the interface shape indicates that in this case, the IBA phase is more wetting,which means that even though the capillary pressure is still the driving force, a drainage typedisplacement occurs in these experiments. This type of displacement also explains why thecomplete removal of IBA from the micromodel was never observed. The values of the contactangles are known to depend on the time of exposure of the solid to both liquids and of themovement of the trapped phase (i.e., being either the advancing or receding phase). Anotherpossible explanation for this different type of displacement could be the different type of glassused for the fabrication of the micromodel. The trapping of the wetting phase in a drainagetype displacement has also been previously mentioned by Lenormand et al. (1983); Chatzis

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and Dullien (1983) and Jamaloei et al. (2010b). The interfaces shapes depicted in Fig. 13are very similar to the ones described by Lenormand et al. (1983). Note that the shape of thechannels (in the paper of Lenormand et al. (1983) the shape of the channels was rectangularand in our case the shape corresponds to a near-circular profile) does not affect the shapes ofthe interfaces neither in throats or pores.

4 Conclusions

We performed microscale experimental observations to study the dissolution behaviour ofmiscible liquid/liquid interfaces in homogeneous porous media, which extends our previouswork on dissolution of binary mixtures within capillary tubes (Stevar and Vorobev 2012)to a homogeneous glass-etched micromodel. Diffusion is traditionally considered to be awell-studied process, which however does not apply to diffusion of liquid/liquid interfaces,when random walks of molecules are strongly affected by intermolecular interactions.

For glycerol/water and soybean oil/hexane (i.e., completely miscible) binary mixtures wefound that the dissolution process was governed by diffusion. However, this is non-Fickiandiffusion with the growth of the volume occupied by the solvent being proportional to t0.3 inthe initial stage and t0.6 in the second stage which lasted for almost the entire duration of theexperiments. There are some similarities between the results reported in this work and theones reported in Stevar and Vorobev (2012) which, in particular, allow us to conclude that theinitial stage of the observed dissolution process is possibly driven by the barodiffusion effect.During the second stage, the solute/solvent interface is very diffusive and the observed timedependence is closer to the one resulting from the Fickian law. Nevertheless, the observedtime dependence is still clearly different from the classical t0.5 power law typical for adiffusive process (Crank 1976).

In the case of IBA/water binary mixture our results are rather qualitative. The experimentsperformed at undercritical temperatures revealed that, despite the fact that the mixture wasout of its thermodynamic equilibrium, diffusive mass transport was negligible. The motionof numerous interfaces within the micromodel obviously caused by capillary forces could benoticed, but the complete removal of IBA from the micromodel was never observed.

To conclude, we note that the results of the current work can be utilized for the developmentof accurate macroscale models for miscible multiphase flows within porous media. To this day,the dissolution of miscible mixtures is still being modelled in a simplistic manner, neglectingthe barodiffusion and capillary effects and modelling the interfacial mass transport on thebasis of the classical Fick law (e.g., Gaponenko and Shevtsova 2010). A micromodel with aregular flow-path pattern was chosen in order to make the results obtained easier employableby the researchers carrying out theoretical and computational modelling.

Acknowledgments This work is done within the framework of the EPSRC project EP/G014337 ‘Pore-levelNetwork Modelling of the Miscible Displacement’. The authors thank Dr. J. Shrimpton and Dr. Dehao Ju forhelping with the optical measurements and Dolomite Centre Ltd., UK for manufacturing the micromodel.

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Dissolution Dynamics of Liquid/Liquid Binary Mixtures