Effects of partial miscibility on drop-wall and drop-drop interactions

Transcription

1 Effets of partial misibility on drop-wall and drop-drop interations C. Tufano, G. W. M. Peters, H. E. H. Meijer, and P. D. Anderson a) Materials Tehnology, Eindhoven University of Tehnology, P.O. Bo 513, 56 MB Eindhoven, The Netherlands (Reeived 4 June 29; final revision reeived 17 September 29; published 26 January 21 Synopsis The effet of the mutual diffusion of two polymeri phases on the interation and oalesene of two nearby drops in quiesent onditions is investigated for two partially misible systems, differing in the misibility of the omponents. Transient interfaial tension measurements show that the polybutene PB /polydimethylsiloane PDMS system is highly diffusive in terms of diffusing low-moleular weight speies, while the polybutadiene PBD /PDMS system is less misible. Drops of the highly diffusive PB/PDMS system at distanes loser than their equivalent radius attrat eah other and oalese with a rate that, in the last stage of the oalesene proess, is the same for all drop ombinations. For the slightly diffusive PBD/PDMS system, no oalesene ours, and, in ontrast, repulsion between the drops is observed. These phenomena are qualitatively eplained in terms of the overlap of diffuse layers formed at the drop surfaes of two, lose enough drops, yielding onentration gradients that ause gradients in the interfaial tension. These gradients yield Marangoni stresses that indue flow leading either to attration or repulsion. To determine whether Marangoni stresses are strong enough to displae a drop in quiesent onditions, single drops of PB and PBD are plaed in a PDMS matri in the viinity of a wall. A lateral drop motion toward the wall is observed for the highly diffusive PB/PDMS system only, while PBD drops do not move. The diffuse-interfae model is onsidered as a good andidate to apture these phenomena desribed sine it ouples the mutual diffusion of the low-moleular weight omponent with both drop and matri, while inluding hydrodynami fores. The presented numerial simulations indeed show a diffusion-indued marosopi motion that qualitatively reprodues the eperimental phenomena observed and support our interpretations. 21 The Soiety of Rheology. DOI: / I. INTRODUCTION Interfaial properties in multi-phase systems are important beause they affet break-up and oalesene events during the proessing of blends, defining the transient and steady morphologies, whih result from the dynami equilibrium between these phenomena. Usually, polymers are onsidered fully immisible e.g., Jones and Rihards 1999 ; Fortelny and Kovar 1988 ; Lyu et al. 2 ; Elmendorp and van der Vegt 1986 ; Rusu and Peuvrel Disdier 1999 ; Verdier and Brizard 22 ; Vinkier et al ; however, ases eist where diffusion of low-moleular weight speies aross an a Author to whom orrespondene should be addressed; eletroni mail: p.d.anderson@tue.nl 21 by The Soiety of Rheology, In. J. Rheol. 54 1, January/February /21/54 1 /159/25/$3. 159

2 16 TUFANO et al. interfae ours in the eperimental time sale Shi et al. 24 ; Peters et al. 25 ; Tufano et al. 28a, 28b, and where mass transfer between the two phases, although limited, reates a gradient in the onentration of migrating moleules, hanging the interfae properties, thus, affeting drop deformation dynamis and film drainage between two approahing drops Klaseboer et al. 2 ; Chevaillier et al. 26. Hu et al. 2 reported that a redution of 3% in the interfaial tension redues the ritial apillary number for oalesene by a fator of 6. Mutual misibility an ause gradients in interfaial tension sine the low-moleular weight fration aumulated at the interfae behaves as a surfatant. To balane this gradient, tangential stresses appear at the drop interfae influening film drainage, usually referred to as Marangoni flow Sriven and Sternling 196 ; Levih and Krylon Depending on the diretion of mass transfer along the drop surfae, gradients in the interfaial tension differ in sign and an aelerate or deelerate film drainage, promoting or suppressing oalesene, respetively. Makay and Mason 1963 showed that the rate of thinning of the film separating two approahing drops inreases when diffusion of a third omponent mutual solvent ours from the drop phase into the matri and dereases in the opposite ase. Pu and Chen 21 and Chen and Pu 21 investigated jump-like oalesene between two aptive drops of oil in water and in the absene of eternal fores, showing that the presene of a third diffusing omponent enhanes oalesene, while the binary oalesene time is retarded by the addition of a surfatant. They proposed an equation to epress the binary oalesene time as a funtion of a so-alled thin-film oeffiient, whih reflets the thin-film properties: the moleular properties of the inner phase, the interfae and the ontinuous phase, thus, the interfaial onentration gradients of surfatant, the visosity of ontinuous phase, and the influene of steri hindrane and temperature on interdiffusion. By interpreting the eperimental data by using this epression, they found support that the larger the differene in drop size, the shorter the oalesene time. Velev et al found thik and very stable aqueous films between oil phases when a surfatant is diffusing from the interfae toward the film and attributed the film stability to the aggregation of surfatant mielles in the film area, generating an osmoti pressure differene between the film interior and the aqueous menisus. Film drainage between two aptive polyethylene oide PEO -water drops in a polydimethylsiloane PDMS matri is found to be very sensitive to an inrease in film radius, and Zdravkov et al. 23 attributed the effet to a depletion of PEO moleules adsorbed on the drop interfaes into the film. The same authors Zdravkov et al. 26 arried out further investigations on the effets of mutual diffusion on film drainage, showing that for highly diffusive systems, the drainage rate is 1 times faster than predited by eisting theoretial models, while, when a slightly diffusive system is onsidered, good agreement with the partially mobile model predition is found. The results are eplained in terms of Marangoni onvetion flows, whih promote film drainage when the overlap of the diffusion layers formed around the drop surfae ours, and slow it down in the opposite ase. Etensive numerial studies have been arried out to desribe the interfae between two or more liquids defined as a spae in whih a rapid but smooth transition of physial quantities between the bulk fluid values ours. Equilibrium thermodynamis of interfaes was developed by Poisson 1831, Mawell 1876, and Gibbs Rayleigh 1892 and van der Waals 1979 developed a model to desribe a diffuse-interfae based on gradient theories that predit the interfae thikness; thus, an interphase, whih tends to beome infinite one the ritial temperature is approahed. A review on diffuseinterfae methods in fluid mehanis was given by Anderson et al Lamorgese

3 EFFECTS OF MISCIBILITY ON DROP INTERACTIONS 161 TABLE I. Seleted model omponents. Sample M n a g/mol M w /M n a g/m 3 mn/m Pa s R mm R 4h m PB/PDMS 635/ / / / PBD/PDMS 8/ / / / a Provided by the supplier. and Mauri 26 used a similar approah to simulate the miing proess of a quiesent binary miture that is instantaneously brought from the two to the one-phase region of its phase diagram. In this work, a highly diffusive system, polybutene PB in PDMS, and a slightly diffusive system, polybutadiene PBD in PDMS, are used to investigate the effets of partial misibility between the two polymeri phases i transient interfaial tension, ii the lateral motion of a single drop, and iii oalesene of two drops in quiesent onditions. Details on the eperimental approah of the transient interfaial tension are given in Tufano et al. 28b. The results are ompared to preditions of the diffuse-interfae model modified to aount for three omponents: the drop phase, the low-moleular weight fration of migrating moleules, and the matri phase. Drop oalesene under quiesent onditions is eperimentally investigated for different drop radii and distanes between them. The results are interpreted in terms of diffusivity of low-moleular weight omponent, in drop and matri phases, whih indue gradients in interfaial tension on the drop surfae. In addition, we show that similar phenomena an be observed for a drop lose to a wall. II. MATERIALS AND METHODS A. Materials The polymers used as dispersed phase are PB Indopol H-25, BP Chemials, UK and PBD Rion 134, Sartomer. The ontinuous phase is PDMS UCT. The materials are liquid and transparent at room temperature. The number average moleular weights M n and the polydispersities M w /M n of the materials are given in Table I. Densities measured with a digital density meter DMA 5, Anton Paar and steady interfaial tension measured with a pendent drop apparatus, at room temperature, are also listed in Table I. Zero shear visosities are measured using a rotational rheometer Rheometris, ARES equipped with a parallel-plate geometry, applying steady shear. All polymers ehibit Newtonian behavior in the range of shear rates applied.1 1 s 1 and the visosities at room temperature are added to Table I. While measuring the interfaial tension, also the hanges in the droplet radii are measured. Variations in the drop radius in 4 h R 4h are used as a measure for the blend diffusivity see the last olumn in Table I. Assuming that the thikness of the diffusion layer around the drop is proportional to R 4h, the system PB/PDMS is more diffusive ompared to the system PBD/PDMS and will have a thik diffusive layer, while the PBD/PDMS system will have a thin diffuse layer. B. Eperimental methods Coalesene eperiments are performed in a home-made Couette devie, ensuring simple shear flow between the onentri ylinders. The diameters of the inner and outer ylinders are m and m, respetively. The ylinders are atuated by two d motors Maon and an rotate independently in both diretions. The motors are

4 162 TUFANO et al. ontrolled by a TUeDAC, an in-house developed digital analog onverter, and two amplifiers are used to strengthen the signals. The real time ontrol of the motors is guaranteed by a home-made software. Images are aquired via a 45 oriented polished surfae plaed below the ylinders. A stereo-mirosope Olympus and a digital amera serve the aquisition of images, whih are further analyzed. In all eperiments, a single droplet is introdued in the stagnation plane obtained by onentri ounter-rotating onditions, using a syringe, and the ritial apillary number is reahed to break the droplet in two or more daughter droplets. The angular veloities of the two ylinders are ontrolled suh that the droplet position is stationary and images an be aquired during the whole proess. When droplets of the required sizes are reated, the flow is reversed, bringing the droplets at the required distane. The reversed flow is hosen to be very slow to avoid any a priori drop deformation. One the flow is stopped, quiesent oalesene is investigated. To further investigate the drop motion indued by gradients in interfaial tension, a ubi ell with flat glass surfaes is used. Drops of PB and PBD are plaed in the ell filled with PDMS, in the proimity of the wall. Images are aquired from the bottom of the ell following the same proedure as desribed for the Couette devie. In order to elude wetting effets on the possible lateral migration of the drops, the same eperiments are arried out in a ubi ell with polytetrafluorethylene PTFE Teflon side walls. III. DIFFUSE-INTERFACE MODEL Diffuse-interfae modeling allows us to aount for interfaes with nonzero thikness. The method is based on the van der Waals s approah of the interfae problem van der Waals 1979 and developed by Cahn and Hilliard The interfae thikness is not epliitly presribed but follows from the governing equations that ouple the thermodynami and hydrodynami fores in the interfae. The main elements of the theory and the oupling of thermodynamis and hydrodynamis are summarized by Anderson et al In this work, the diffuse-interfae model is applied to desribe a three-phase systems, onsisting of a low-moleular weight fration, a drop phase, and a matri. First, the governing equations are summarized, the numerial methods are outlined, and oneand two-dimensional results are presented and ompared with eperimental observations. A. Governing equations For a hemial inert N-omponent system, the mass balane an be written as + v =, t 1 with as the density of the miture, defined as the sum of the N-omponent densities, = N i=1 i, and v is the baryentri veloity, N v = 1 i v i, i=1 2 where i, v i are the density and veloity of the ith omponent, respetively. The omposition equation reads as

5 EFFECTS OF MISCIBILITY ON DROP INTERACTIONS 163 i t + iv = j i, 3 where j i is the mass flu of the ith omponent of the onsidered fluid, with i ranging between 1 and N. The momentum balane, taking into aount the mass balane Eq. 1, an be written as v t + v v = fe +, 4 where f e denotes eternal fores suh as gravity, is the Cauhy stress tensor. To omplete this set of equations, onstitutive relations are required for the Cauhy stress tensor. Different from lassial thermodynamis, in whih the internal energy u is a funtion of s, the speifi entropy, and the density of the omponents, u=u s,, in Cahn Hilliard ase an etra non-loal term is introdued to desribe inhomogeneous fluids, and the internal energy is defined as u=u s,,. Applying gradient theory, the Cauhy stress tensor is given by see Lowengrub and Truskinovsky 1998 N u = pi i, i=1 i 5 where p is the pressure, is the etra stress tensor that, assuming isothermal onditions, for a Newtonian system is taken to be =2 D, D is the deformation tensor given by D= v+ v T /2, and is the visosity of the miture for a more omplete desription of the model, see Khatavkar et al. 27a and Prusty et al. 27. An additional gradient term, similar to the Cauhy stress tensor, is added to the hemial potential of eah of the N speies, and to the pressure, i = i u i, 6 N pi = p I i u, i=1 i 7 in whih i and p are defined with respet to the homogeneous referene state. Substituting the epression of and in the momentum balane, and writing i as i, with i = i /, the momentum balane reads as v t + v v = fe + 2 D p N i=1 u i i. 8 It was shown by Lowengrub and Truskinovsky 1998 that N i=1 N 1 u i = i f+ i N i, i=1 where f=u Ts is the speifi Helmholtz free energy of the system, T is the temperature, and s is the entropy. Substituting Eq. 9 and the speifi Gibbs free energy defined as g=f+p/, ineq. 8, and dividing all terms by, the momentum balane redues to 9

6 164 TUFANO et al. v t + v v = fe D g+ i N i. 1 i=1 The g term an be onsidered as a modified pressure and the interfaial tension is now evaluated as a body fore N 1 i=1 i N i, as shown by Lowengrub and Truskinovsky If we substitute the density of eah omponent i with its mass fration i = i /, and we epress j i in terms of i, Eq. 3 an be written as N 1 i t + v i = M i. 11 Finally, writing the internal energy u in terms of the speifi Helmholtz free energy of the system f, the hemial potential follows: i N = f 1 f i i. 12 B. Governing equations for a three-phase system When a three-phase non-homogeneous system is investigated assuming isothermal onditions, density-mathed phases, inompressible fluids, onstant visosities of the phases and negleting inertia and eternal fores, and using 1 and 2 in the momentum balane to epresses the hemial potential differenes with respet to the third omponent, the system of governing equations redues to: Mass balane Momentum balane Composition equation v =. = g+ 2 v i t + v i =M i 2 i i=1,2. 15 Chemial potential i N = f i i i=1,2. 16 For this three-phase system, we used = M 17, where and are input parameters in the model, desribing the mobility between low-moleular weight and drop, and matri and drop, respetively. Note that, in general, and an be a funtion of i, but they are taken onstant here. In the Cahn Hilliard theory, the speifi Helmholtz free energy of the system is given by the sum of a homogeneous part and a gradient ontribution

7 EFFECTS OF MISCIBILITY ON DROP INTERACTIONS FIG. 1. Contour plot of the free energy Eq. 2 on the Gibbs triangle for a=b=d=1/4 left and for a=4, b=2, and d=1 right. f, =f , 18 where f is the homogeneous part, and the non-loal term of the free energy, and is the gradient energy parameter assumed to be onstant, i.e., = I, where we neglet any possible off-diagonal terms. The model requires one more equation of state to desribe the free energy in order to solve the system of Eqs C. Ginzburg Landau approimation Based on the lassial Flory Huggins theory, the intensive free energy f per monomer of a three-phase system, at a given temperature and pressure, an be written as f KT = 1 N 1 ln N 2 ln N 3 ln , 19 where K is the Boltzmann onstant, T is the temperature, N i and i with i=1,2,3 are the hain length and volume fration of the three omponents, respetively, and ij the Flory Huggins interation parameter between the omponents i and j. Chosen a threephase system, this interation parameter defines whether miing or demiing ours. For numerial purposes, a Taylor epansion of Eq. 19 around the ritial point is used as an approimation and as suh the system of equations is orret only in the viinity of the ritial point, and therefore, when applied to the present, far-fromritiality ase, only provides a qualitative piture of the phenomenon. In Kim et al. 24, the approimation for the free-energy formulation proposed for a three-phase system reads as f =f 1, 2 = 1 4 a b d 1 2 3, 2 where 3 =1 1 2, and a, b, and d are onstants. When these three onstants are assumed equal to 1/4, as proposed by Kim et al. 24, the surfae plot of the free energy for the ternary system presents free-energy minima in the orners of the diagram and at the enter of it, where the system is fully misible see Fig. 1, left. We will use this epression to validate our numerial ode by omparing with results from Kim et al. 24. Our system onsists of low-moleular weight speies partially misible with the drop and matri phases, while drop and matri are immisible. This requires different values of the parameters in Eq. 2 or another free-energy epression sine by just hanging the parameter values a=4, b=2, and d=1 see Fig. 1, right, it an be ob-

8 166 TUFANO et al FIG. 2. Surfae plot left and ontour plot right of the free energy Eq. 21 on the Gibbs triangle. served that the free-energy ontour plot remains symmetrial around one of the aes. Changing the parameter values therefore does not solve the problem, and we use an alternative free-energy formulation for ternary partially misible systems proposed by Kim and Lowengrub 25, f 1, 2 = a b 2 d 2 + e l Using values of a=2, b=.2, d=.2, e=1.2, and l=.4, the surfae and ontour plot of this free-energy formulation are shown in Fig. 2. The ontour plot orresponding to Eq. 21 is no longer symmetri; therefore, it is suitable to desribe our three-omponent system. D. Numerial method The resulting system that needs to be solved, i.e., Eqs , is non-linear and time-dependent. For the temporal disretization, a first-order Euler impliit sheme is used. The non-linear term in the hemial potential equation is linearized by a standard Piard method in eah time step. Instead of substituting the hemial potential in the omposition equation, it is treated as a separate unknown. The main advantage of this approah is that only seond-order derivatives need to be evaluated. So within eah yle of a time step, the hemial potential and onentration are solved together, using the veloity from the previous time. The veloity and pressure are determined by using the omposition and hemial potential from the previous time step. Roughly within five iterations, a solution is found for the non-linear problem of eah time step. More details about the iteration sheme an be found in Keestra et al. 23 and Khatavkar et al. 26. A seond-order finite element method is used for spatial disretization of the set of equations. The flow problem is solved using the veloity-pressure formulation and disretized by a standard Galerkin finite element method. The effet of the interfae is inluded as a known volume soure term. Taylor Hood quadrilateral elements with ontinuous pressure that employ a biquadrati approimation for the veloity and a bilinear approimation for the pressure are used. The resulting disretized seond-order linear algebrai equation is solved using a diret method based on a sparse multifrontal variant of Gaussian elimination HSL/MA41, Amestoy and Duff 1989 ; Amestoy and Puglisi 22. E. Validation of the model To validate the ternary diffuse-interfae model and its implementation, a test ase as desribed by Kim et al. 24 is simulated. Here, a ternary system in a one-dimensional domain is defined with the free-energy formulation of Eq. 2. The one-dimensional

9 EFFECTS OF MISCIBILITY ON DROP INTERACTIONS 167 FIG. 3. Time evolution of the onentrations 1, 2, and 3 at three different dimensionless times, i.e., t=, 5, and 1. domain with dimensionless length 1 is defined ranging from to 1. The domain is disretized with 2 s-order elements yielding 41 nodes and the uniform time step is t= The simulations are started with the following initial onditions: i = os os 5 i=1,2, 3 =1 1 2, where 1. The resulting time evolutions of the three-omponent onentrations are shown in Fig. 3 at three different times, i.e., t=, 5, and 1. The obtained results fully agree with those reported as in Kim et al. 24. This validated model is used hereafter to ompute and interpret the eperimental results. F. Influene of the mobility parameters The ternary diffuse-interfae model as desribed in Se. III B needs five material parameters, i.e.,,,,, and, one the hoie for the homogeneous part of the free energy has been defined. In general, these parameters are not known or easily obtained for a given polymeri system and, therefore here, we will restrit ourselves to a qualitative analysis. In this paper, the main objetive of the use of the ternary diffuse-interfae model is to be able to distinguish the epeted diffusion-indued flow behavior of the highly diffusive PB/PDMS system and the slightly diffusive PBD/PDMS system and ompare with our eperimental observations. Sine for these polymeri systems the material parameters and ternary phase diagram are unknown, Eqs are solved in their dimensionless form. For a large range of the parameters, one-dimensional simulations were arried out, where purely diffusion is onsidered, to obtain insight in the relevant magnitude of the parameters and their effet on the diffusion proess of the drops. The knowledge obtained from this eerise then serves as input for the two-dimensional simulations presented and disussed further on. The free-energy epression adopted is given in Eq. 21, and the gradient-energy parameter is fied to The visosity and density are both set equal to 1. The influene of hanges in the mobility parameters low-moleular weight-drop and matri-drop is now investigated. The one-dimensional domain with dimensionless length 1 is defined ranging from to 1. The domain is disretized with 2 s-order elements yielding 41 nodes. The drop is plaed in the middle, with an initial diameter equal to the 5% of the domain. The onentration of the low-moleular weight omponent is taken to be 3% and time steps of t=1 1 4 are used. A shemati piture of the domain and the initial onentration is shown in Fig

10 168 TUFANO et al. FIG. 4. Shemati representation of the initial onditions. The transient onentration profiles of the low-moleular weight speies, drop and matri, are shown in Fig. 5 for =1 1 2 and in Fig. 6 for =9 1 2, respetively. In both ases, is taken equal to It is seen that the shapes of the onentration FIG. 5. Conentration profiles in time for low-moleular weight omponent top left, drop top right, and matri bottom for a.3 low-moleular weight onentrated blend. Mobility parameters: =1 1 2 and =4 1 4.

11 EFFECTS OF MISCIBILITY ON DROP INTERACTIONS 169 FIG. 6. As in Fig. 5, now with: =9 1 2 and =4 1 4 ase A. profiles do not hange muh, but the time sale of reahing a ertain profile hanges onsiderably. For suh a system, the interfaial tension an be related to the omposition as 1 d 2 LMW d. 23 d = The resulting interfaial tension behavior is shown in Fig. 7 left when hanging gradually, in the middle when hanging and on the right when lowering by orders of magnitude. For these parameter sets, a lear minimum in interfaial tension is observed that is reahed at shorter times for a gradually inreasing value, and it is not hanging its value. Changes in affet the transient behavior of the system in the way it reahes the steady state. When is inreased gradually, the time at whih the minimum value in interfaial tension is reahed stays the same but its value redues. When is lowered by orders of magnitude Fig. 7, right, the minimum disappears and the interfaial tension dereases only in a muh slower way. The ases when a minimum appears are representative of highly diffusive systems, when the minimum disappears the systems behave as a slightly diffusive one. In onlusion, diffusivity an be ontrolled by. G. Influene of the low-moleular weight fration To show how hanges in the initial onentration of low-moleular weight speies affet the interfaial tension of the system, we hoose the ase with a lear minimum =9 1 2 and =4 1 4, whih we will address as ase A, and three different

12 17 TUFANO et al. FIG. 7. Interfaial tension omputed with Eq. 23 for a.3 low-moleular weight onentrated blend. The influene of gradual hanges in the mobility parameters top left and top right when they differ two orders of magnitude and when reduing this differene bottom are shown. FIG. 8. LMW onentration profiles when =9 1 2 and =4 1 4 ase A for initial onentrations of LMW speies of.3 top left,.1 top right, and.1 bottom.

13 EFFECTS OF MISCIBILITY ON DROP INTERACTIONS 171 FIG. 9. Transient interfaial tension eperiments left and numerial preditions right. initial onentrations of low-moleular weight omponent, respetively, 3%, 1%, and 1% are investigated. The same drop size and time steps are used as in the previous ase. The onentration profiles along the drop diameter at different time steps are shown in Fig. 8. Reduing the initial onentration of migrating moleules redues the typial radial length sale over whih diffusion is observed and it shortens the time needed to omplete the diffusion proess. We now have sets of interfaial tension evolutions that an be ompared with our eperimental results. H. Interfaial tension results Measured transient interfaial tensions are shown for the two systems used in Fig. 9 left. For the highly diffusive system, the interfaial tension dereases first, orresponding to thikening of the interphase, followed by an inrease attributed to depletion where after it reahes a plateau value in the late stages. The slightly diffusive system shows only thinning i.e., an inrease in the interfaial tension thikness before the plateau value is approahed Tufano et al. 28b. InTufano et al. 28b, the differenes in the interfaial behavior of these two systems are partially attributed to their different polydispersities. The PB has higher polydispersity ompared to the PBD, i.e., in the system PB/PDMS a larger amount of moleules will partiipate to the diffusion proess ompared to the PBD/PDMS system. Based on that, our first approah is to model the two systems by using the three-phase diffuse-interfae model desribed in Se. III, with the free-energy formulation reported in Eq. 21. The mobility parameters are hosen to be =9 1 2 and =4 1 4 ase A, and the parameter = To distinguish the two ases, the highly diffusive system is simulated imposing 3% low-moleular weight LMW onentration, while, for the slightly diffusive ase, this onentration is set equal to 1%. The omputed behavior of interfaial tension in time in the two ases is shown in Fig. 9 right. Given all ompleities at hand, the model desribes the eperimentally observed behavior qualitatively quite well. We remark again that the results from the simulations are dimensionless in ontrast to the presented eperimental transient interfaial tension for the two polymeri systems; apparently, the time sales assoiated with the diffusion of the drops are on the order of 1. Within a dimensionless time of 1, we observe in Fig. 9 right that the omputed interfaial tension shows a similar transient behavior as in the eperiments and that after about a dimensionless time of 1.5 a steady value of is reahed.

14 172 TUFANO et al. FIG. 1. Mutual attration and oalesene between two drops of PB in PDMS. Radii are 341 and 313 m, distane at the time t= s is 11 m. Residene time at t= s and t r =18 s. IV. EXPERIMENTAL RESULTS A. Drop-drop interation: PB/PDMS system Figure 1 shows an eample of two drops of PB with nearly the same diameter and at a distane where, normally in a sharp interfae regime, no interation is epeted. The initial distane between the drops is reated by applying a weak flow during drop approah, ensuring that the drops keep their spherial shape and that there is no influene of flow on oalesene. One the desired distane d between the drops is reahed, the flow is stopped and no other eternal fores are applied denoted as time t= s in Fig. 1. At t= s, the residene time of the drops in the matri is t r =3 min and it is learly seen that the drops attrat eah other, the film is drained, and when rupture ours the drops oalese. Figure 11 shows an eample of mutual attration of drops with different radii and with a shorter residene time at t= s t r =4 min. A large number of suh eperiments are arried out, using drops of different sizes, plaed at different distanes and having different residene times. Drops of PB always attrat eah other when they are at a distane less than an order of the drop radius. In order to ompare the results obtained with different drop size, the ratio distane over equivalent radius is onsidered. The equivalent radius R eq is defined as 2 R eq = 1 R R 2, 24 where R 1 and R 2 are the radii of the two drops. Figure 12 shows the time evolution of the distane between the drops for all eperiments. In the inset plot, all urves are horizontally shifted to the most right urve to ompensate for differenes in their initial distane. In the final stages of the measurements, approimately the last 1 s before oalesene ours, the drops all attrat with the same rate. Aging effets are also investigated monitoring drops with different resident times. In the first 3 min after the introdution of the drops in the matri, diffusion is in progress and the interfaial tension redues due to thikening of the interfae. However, as shown in Fig. 12, no serious variations in the rate of attration in the late stages before oalesene ours ould be deteted. Zdravkov et al. 26 reported for a similar PB/PDMS blend that film drainage is approimately 1 times faster ompared to the partially mobile model preditions. They attributed the high FIG. 11. As in Fig. 1, now radii are 22 and 275 m, distane at the time t= s is 95 m. Residene time at t= s and t r =24 s.

15 EFFECTS OF MISCIBILITY ON DROP INTERACTIONS 173 FIG. 12. Time evolution of the saled drop distane d/r eq for the PB/PDMS system. Inset plot: urves shifted to the most right eperimental urve. drainage rate to Marangoni flow, ating in the same diretion as the film drainage flow. Note that in the diffuse-interfae model, the Marangoni stresses are the marosopi manifestation of the Korteweg stresses. This high drainage rate is onfirmed in the eperiments presented here. In onlusion, for highly diffusive systems, mutual diffusion annot be negleted sine it has an overruling effet on drop oalesene and, therefore, it plays a ruial role in the morphology evolution of polymer blends, as also shown by Tufano et al. 28b for diluted blends of PB/PDMS. B. Drop-drop interation: PBD/PDMS system The system PBD/PDMS shows a ontinuous inreasing interfaial tension related to a thin diffusive layer, whih approah a plateau value faster than the PB/PDMS system see Fig. 9. When two PBD drops are brought in lose ontat and left in quiesent onditions, it is observed that they repel eah other see Fig. 13. In all the ases investigated, i.e., drops with different sizes and at different distanes, repulsion is observed. For a similar PBD/PDMS blend, Zdravkov et al. 26 reported that the film drainage slows down with time and eventually reverses. This was again attributed in that ase to Marangoni stresses, whih may ause reversal of the film drainage and eplain the repulsion observed for this system. FIG. 13. Mutual repulsion between two drops of PBD in PDMS. The radii are 597 and 572 m.

16 174 TUFANO et al. When diffusion ours from the drop into the matri phase, a gradient in the onentration of the migrating moleules is generated in the radial diretion forming a diffuse layer that develops around the drops. The more diffusive the material, the thiker this layer. When two drops approah, the thikness of this layer determines whether and when overlap ours. In the spae where two layers overlap, a higher onentration of migrating moleules is found ompared to their onentration on the remaining of the drop surfae. Gradients in moleule onentration result in a gradient in interfaial tension, and in the overlap zone a lower interfaial tension is found. This gradient in interfaial tension generates tangential Marangoni stresses along the drop surfae that at to redue the interfaial tension gradient. Therefore, they indue a onvetive flow from the zone of overlap to the sides. This flow ats in the same diretion of the film drainage, aelerating this proess, and, therefore, attration between drops ours and oalesene is promoted. The system PB/PDMS is an eample of suh a diffusive system. When the material presents a thin diffuse layer, the overlap does not our. The film drainage will then drag moleules from the zone between the drops to the sides. It generates the opposite onentration gradients, and, therefore, opposite interfaial tension gradients. The tangential Marangoni stresses at in this ase in the diretion opposite to the film drainage, retarding oalesene. While thinning of the film between the drops ours, the drainage rate redues. It an happen that the onvetive flow indued by Marangoni stresses overrules film drainage, reversing the thinning rate. The matri flows bak between the drops that move further a part and the eperimentally observed repulsion ours. To support the idea that Marangoni stresses an indue lateral migration of drops and, eventually, enhaned oalesene when highly diffusive systems are used, single-drop measurements are performed. Single PB and PBD drops are reated in the viinity of a flat wall and their possible motion, attributed to diffusion, is reorded. To elude any effet due to wetting, glass and PTFE walls are used. C. Drop-wall interation: PB/PDMS system When a PB drop is plaed in the matri, diffusion of short moleules from the drop into the matri ours. If the drop is lose enough to a wall d R, there is less spae for migration of the shorter moleules on the wall side and, as a onsequene, their onentration will be larger than on the rest of the drop surfae. This indued gradient in onentration, i.e., in interfaial tension, along the drop surfae, generates Marangoni flows, whih will at as to balane the onentration gradient. Movement of the migrating moleules aumulated between the drop and the wall, toward the sides of the drop, will also drag moleules of the matri. The immediate onsequene is the thinning of the matri film between the drop and the wall. The drop then moves toward the wall, touhes it, and eventually wets it see Fig. 14, where a glass wall is used, and Fig. 15, where a Teflon wall is used. FIG. 14. Lateral PB-drops migration toward a glass wall. The diameters are 58 and 333 m. The initial distanes from the wall are 163 and 59 m, respetively. The solid line represents the wall.

17 EFFECTS OF MISCIBILITY ON DROP INTERACTIONS 175 FIG. 15. Lateral PB-drops migration toward a Teflon wall. The diameter is 247 m and the initial distane from the wall is 45 m. Line represents the wall. D. Drop-wall interation: PBD/PDMS system When a drop of PBD is plaed lose to a wall, no lateral migration is found eperimentally. The time sale of the eperiment is limited by the vertial movement of the drop, due to the differene in density. The aquisition is stopped when the drops start to move out of fous. In Figs. 16 a and 16 b, a drop of PBD is plaed lose to a glass and a Teflon wall, respetively. In the time sale investigated, no lateral movement is seen. V. COMPUTATIONAL RESULTS In the previous setions, the desribed eperimental results demonstrated the different diffusion-indued flow behavior for the two different polymeri systems, i.e., the drops are attrated and may oalesene, or repulsion is observed. Earlier, we saw that the ternary diffuse-interfae method desribes the transient behavior of the interfaial tension in a qualitative sense and that parameters for the mobilities and surfae interation parameter are defined. Now the full model inluding hydrodynami interations is applied, using the parameters from the one-dimensional model, to simulate the two drops in lose proimity. Figure 17 gives a shemati representation of the two ases onsidered here. First, we deal with drop-drop interation; seond, with drop-wall interation. For the drop-wall simulations, the number of elements used in the and y diretion are N =N y =6, while for the drop-drop simulations N =16 and N y =8. If we indiate with v= v,v y, the following boundary onditions are defined. For the drop-drop ase homogeneous Dirihlet for the veloity and homogeneous Neumann boundary onditions for the onentration and hemial potential are presribed FIG. 16. No lateral PBD-drops migration toward a wall is found. a Glass wall, d drop =65 m; residene time 6 s. b Teflon wall, d drop =9 m; residene time 72 s. The solid line represents the wall.

18 176 TUFANO et al. FIG. 17. Shemati representation of the omputational domain used for the drop-drop left and drop-wall right simulations. v = i n = i n = at j j = 1,2,3,4. 25 For the drop-wall ase, similar boundary onditions are presribed eept for the solid wall.3.25 LMW M =9*1 2 1 M =4*1 4 2 t= t=1*1 4 t=3*1 4 t=1* Drop =9*1 2 =4*1 4 t= t=1*1 4 t=3*1 4 t=1* Matri =9*1 2 =4*1 4 t= t=1*1 4 t=3*1 4 t=1* FIG. 18. Conentration profiles in time for LMW omponent top left, drop top right, and matri bottom for a 3% LMW onentrated blend. Mobility parameters: =9 1 2 and =4 1 4 ase A.

19 EFFECTS OF MISCIBILITY ON DROP INTERACTIONS LMW =4*1 4 =4*1 4 t= t=1*1 4 t=5*1 4 t=2* Drop =4*1 4 =4*1 4 t= t=1*1 4 t=5*1 4 t=2* Matri =4*1 4 =4*1 4 t= t=1*1 4 t=5*1 4 t=2* FIG. 19. Conentration profiles in time for LMW omponent top left, drop top right, and matri bottom for a 3% in LMW onentrated blend. Mobility parameters: =4 1 4 and =4 1 4 ase B. v = i n = i n = at 4, 26 v i n = v y n = = n = at j j = 1,2,3. 27 i n Note that the subsript i in above boundary onditions refers to the two different phases. The initial veloity is set to zero at the start of the simulation, where the initial onentrations 1 and 2 are spatially dependent and define the loation of the drop, similar as in the one-dimensional simulations. A. Drop-drop interation First, the numerial results for the drop-drop interation ase are disussed. For the two ases A3%, B3%,, the onentration profiles over the drop-drop enter line are shown in Figs. 18 and 19 at four different harateristi times. Results are presented for 1, 2, and 3 = For ase A, the mobility parameters are =9 1 2 and =4 1 4, while, for ase B, = = Clearly for the A3% system in Fig. 18, the highly diffusive system, large variations in the onentrations are observed and the

20 178 TUFANO et al FIG. 2. Top row: -omponent veloity for A3 at times t=.5 and.8; the drops approah eah other. Bottom row: -omponent veloity for A3 at times t=.1 and.2; the drops slowly move apart. two drops merge after suffiient time. Fousing on the onentration, it is seen that the drop-drop attration is present for the A3%; although not shown here, but this is even the ase for an inreased drop-drop distane. For the B3% system, the drops appear to remain stationary. To determine without doubt if the two drops are really moving by attration or repulsion, the indued veloity field is studied in more detail. In Fig. 2, the omponent of the veloity is shown for the A3% and B3% ases at two different harateristi times. Isoline of the veloity in the figures denotes the regions of positive and negative veloities. The results from the diffuse-interfae model apture the trends as shown in the eperiments remarkably well; i.e., for the A3% ase, we see that the left drop has a positive veloity, while the drop on the right has a negative veloity. In other words, the drops attrat eah other. For the B3% ase, an opposite veloity diretion is observed; thus, the drops repulse. Similar as observed in the eperiments, the time sale for attration is muh faster than for repulsion. The separation between the omponents stays learly present in the range of time, and for the two drop-drop distanes, investigated for the B-ases. These results indiate that suffiient low-moleular weight speies that an diffuse fast enough into the matri A3% versus B3% are needed to ativate the drop-drop attration. In addition, they onfirm the idea that onentration-gradient indued Marangoni stresses promote the drainage of the film between two droplets in ase of a highly diffusive system.

21 EFFECTS OF MISCIBILITY ON DROP INTERACTIONS LMW =9*1 2 =4*1 4 t= t=1*1 4 t=2*1 4 t=3*1 4 t=5*1 4 t=1* Drop =9*1 2 =4*1 4 t= t=1*1 4 t=2*1 4 t=3*1 4 t=5*1 4 t=1* Matri M =9*1 2 1 =4*1 4 t= t=1*1 4 t=2*1 4 t=3*1 4 t=5*1 4 t=1* FIG. 21. Conentration profiles in time for LMW omponent top left, drop top right, and matri bottom for a 3% LMW onentrated blend. Mobility parameters: and =4 1 4 ase A. B. Drop-wall interation As the A3% and A1% systems have the right time-dependent behavior for the interfaial tension, these two systems seem to be good andidates to investigate the drop-wall interation of a highly diffusive and a slightly diffusive system, respetively. The onentration profiles over a line through the drop enter and perpendiular to the wall are shown in Figs , respetively. The A3% and the B3% ases show, indeed, the behavior that is antiipated from the drop-drop results: a lear interation of the drop with the wall for the A3% ase and a stationary drop for the B3% ase. The A1% ase also shows drop-wall interation, although not as strong as the A3% ase see the onentration levels. This is also aused by the impliitly resulting 9 ontat angle for the drop-wall interation. To generate more realisti results, the modeling should be etended with an etra free-energy funtion at the wall that defines this ontat angle Khatavkar et al. 27b for the drop-drop interation, the ontat angle is not an issue. Note that a variation in the drop-wall distane did not give any new insights and no results are presented here for these ases. From the results presented, it is onluded that the diffusive-interfae model an generate, so far only in a qualitative way given the diffiulties to obtain all parameters for the model, the phenomena observed in the eperiments, even though a number of assumptions are used. The diffuse-interfae method is therefore onsidered as a good andidate to desribe these phenomena. For a more quantitative omparison, as disussed earlier, besides an eperimentally validated free-energy formulation, also input for the mobility, visosity, and non-loal interation are needed.

22 18 TUFANO et al LMW =9*1 2 =4*1 4 t= t=1*1 4 t=2*1 4 t=5* Drop =9*1 2 =4*1 4 t= t=1*1 4 t=5*1 4 t=1* Matri =9*1 2 =4*1 4 t= t=1*1 4 t=2*1 4 t=5* FIG. 22. As Fig. 21, now with 1% LMW onentration. VI. CONCLUSIONS The effets of mutual diffusion on interfaial tension, drop-drop, and drop-wall interations in quiesent onditions are investigated eperimentally and numerially. A highly diffusive system PB/PDMS and a slightly diffusive system PBD/PDMS are used at room temperature. Just after ontat between the phases is made, the transient interfaial tension of the highly diffusive system redues as a onsequene of the low-moleular weight fration migration from the drop into the interphase, yielding the formation of a thik diffuse layer around the drop surfae. While time proeeds, after reahing a minimum, the interfaial tension inreases due to low-moleular weight speies migration from the interphase into the matri, leading to depletion of the diffusive layer. One the diffusion proess is ehausted, a plateau in interfaial tension is reahed and sustained. The slightly diffusive PBD/PDMS system, in ontrast, shows only an inrease in the interfaial tension, orresponding to the migration of the fewer migrating moleules polydispersity is lose to one into the matri followed by leveling off to a plateau value that is higher ompared to the PB/PDMS system, whih is attributed to the higher moleular weight of the drop phase. Drop-drop interation eperiments, arried out with isolated pairs of drops and in quiesent onditions, show that partial misibility affets the final morphology of the system. Drops of the highly diffusive PB/PDMS system attrat and oalese when plaed at initial distanes smaller than their equivalent radius. The rate of attration, in the last 1 s of the eperiments, is the same for a wide range of drop sizes radii ranging

23 EFFECTS OF MISCIBILITY ON DROP INTERACTIONS LMW =4*1 4 =4*1 4 t= t=1*1 4 t=5*1 4 t=1*1 4 t=4* Drop =4*1 4 =4*1 4 t= t=1*1 4 t=5*1 4 t=1*1 4 t=4* Drop =4*1 4 =4* t= t=1*1 4 t=5*1 4 t=1*1 4 t=4* FIG. 23. As Fig. 21, now with mobility parameters =4 1 4 and =4 1 4 ase B. between 9 and 35 m and different initial distanes between them. The attration is eplained in terms of the overlap of the diffusive layers around the drops, yielding gradients in interfaial tension and, thus, Marangoni flows ating in the film drainage diretion, enhaning oalesene. When the slightly diffusive system PBD/PDMS is onsidered, with a thin diffuse-interfae, no attration ours and, when the drops are plaed lose together, repulsion between them is observed. Numerial simulations with a three-omponent diffuse-interfae method predit qualitatively the transient interfaial tension, drop-drop, and drop-wall interations, as observed in the eperiments. However, for a more quantitative omparison, more studies are needed to define an eperimentally validated free-energy formulation and realisti values to use as input parameters for our systems. ACKNOWLEDGMENTS The authors thank Roberto Mauri and Dafne Molin from Università di Pisa, Italy, for fruitful disussions on this subjet and the ontribution of Dafne on the first version of the ternary diffuse-interfae ode. Referenes Amestoy, P. R., and I. S. Duff, Memory management issues in sparse multifrontal methods on multiproessors, Int. J. Superomput. Appl. 7,

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