Effect of the adsorption component of the disjoining pressure on foam film drainage

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1 Collod J. 75 (2013) [arxv ] Effect of the adorpton component of the djonng preure on foam flm dranage Stoyan I. Karakahev 1, Anh V. Nguyen 2 and Roumen Tekov 1 1 Department of Phycal Chemtry, Unverty of Sofa, 1164 Sofa, Bulgara 2 Department of Chemcal Engneerng, Unverty of Queenland, Brbane, QLD 4072, Autrala The preent work tryng to explan a dcrepancy between expermental obervaton of the dranage of foam flm from aqueou oluton of odum dodecyl ulfate (SDS) and the theoretcal DLVO-accomplhed Reynold model. It hown that, due to overlap of the flm adorpton layer, an adorpton component of the djonng preure mportant. The pre-exponental factor of the adorpton component wa obtaned by fttng the expermental dranage curve. It correpond to a lght repulon, whch reduce not only the thnnng velocty a oberved expermentally but correct alo the flm equlbrum thckne. Karakahev et al. 1 have nvetgated the dranage of foam flm of dlute aqueou oluton of odum dodecyl ulfate wthn the concentraton range µm. They tred to decrbe the knetc of foam flm thnnng by the Reynold lubrcaton approxmaton accountng for the Marangon effect, urface hear vcoty and DLVO force. Sgnfcant dcrepancy between the theoretcal predcton and the expermental reult wa oberved. The detaled analy howed that the devaton of the theory from the expermental data orgnate from the nteracton between the flm urface. Therefore, t wa concluded that the clacal DLVO theory only not uffcent to match the expermental data. It wa uggeted that the dcrepancy between theory and experment due to a neglected varaton of the adorpton component of the urface tenon durng the flm dranage. Large number of lterature confrm the applcablty of the DLVO theory to foam flm. However, number of paper 2-6 report devaton of th theory from expermental data. Th dcrepancy pronounced motly n thn flm between hydrophobc urface. To olve the problem ome author 7-11 ntroduced n the theory an addtonal non-dlvo force, the o-called hydrophobc force, whch can be attractve or repulve There number of attempt n the lterature to explan the nature of the hydrophobc nteracton but tll no full agreement of the opnon reached. The clacal DLVO theory doe not account alo for other nteracton n the thn lqud flm. For ntance, the nteracton between the overlappng dffuve adorpton layer hould contrbute to the overall nteracton between the flm urface and th contrbuton hould ncreae wth decreang flm thckne. The dea of the adorpton nteracton between the flm urface orgnate from the work of Ah, Everett and Radke 20 and t further developed by the Ruan chool of collod chemtry. The dperon nteracton between the olute and the flm urface accounted for and t reult n a correcton n the van der Wall component of the djonng preure. Th

2 addtonal adorpton term n the total nteracton between the flm urface could be mportant but t ha been overlooked n a large volume of lterature caung dverty of the opnon regardng the hydrophobc nteracton. The reaon for th that the reearcher cted above have decrbed the urfactant dtrbuton only a a reult of nteracton wth the urface but neglected the nteracton between olute molecule. Of coure, the latter are not mportant n dlute oluton far away from the urface, but when the adorpton condered the concentraton near a urface tremendouly ncreaed. Tekov and Schulze 17 uggeted frt a clear thermodynamc nterpretaton of the adorpton term n the total djonng preure. They called t hydrophobc force, nce the orgn of the adorpton the urface hydrophobcty and the urfactant ablty to reduce t. The am of th paper to employ th approach for explanaton of our expermental data. 24 The good agreement wll certanly draw attenton on the mportance of the adorpton djonng preure. Accordng to the thn lqud flm thermodynamc any change of the flm free energy F at contant temperature gven by df pdv da dn (1) where the extenve flm parameter are volume V, flm area A and number of mole { n } of the flm component. The relatonhp between the ntenve parameter preure p, flm tenon and electrochemcal potental { } gven by the Gbb-Duhem equaton Vdp Ad nd 0 (2) It known that the thn lqud flm are anotropc tructure 25 and ther preure tenor poee two dtnct component, the normal and tangental one. At equlbrum the normal component of the preure tenor equal to the ga preure outde, whle the tangental component equal to the preure n the mencu adjacent to the flm. The preure p the normal component of the preure tenor. The flm tenon cont n two addtve, 26 where h V / A the flm thckne, 2 h (3) The purely nterfacal part twce the flm urface tenon whle the bulk part accounted by the djonng preure. Introducng Eq. (3) n Eq. (1) the latter change to df padh ( p ) hda 2dA dn (4)

3 It obvou now that the normal and tangental component of the flm preure tenor are not equal and the djonng preure ther dfference. Ung Eq. (3) one can derve an alternatve form of Eq. (2) Vdp Ad(2 h) nd 0 (5) After Gbb the flm can be dealzed by fllng t wth the bulk lqud from the mencu. Hence, ubtractng from Eq. (5) the Gbb-Duhem relaton dpl cd for the lqud n the mencu, where { c } are the concentraton of the chemcal component there, and keepng n mnd that p p L, one yeld an mportant nterfacal Gbb-Duhem relaton 27 d d dh /2 (6) where { ( n cv)/ 2 A} are the component adorpton. Eq. (6) provde traghtforward an mportant defnton of the djonng preure a the thckne dervatve of the flm urface tenon 2( ) h (7) a well a the followng Maxwell relaton for the djonng preure ( ) h 2( ) h (8) The latter already hnt the mportant effect of adorpton on the djonng preure. 28,29 Snce the urfactant could be charged pece the flm urface tenon depend on electrotatc a well. It can be plt nto uperpoton of water, electrotatc and adorpton component, W EL, whch are ndependent f the urface potental doe not depend on the flm thckne. Thu, durng the flm dranage the adorpton and urface charge denty, repectvely, can vary but the electrotatc component wll not be affected by. At contant temperature the water component depend only on the flm thckne, whle the urfactant component depend on the adorpton. Subttutng th preentaton n Eq. (7) the djonng preure plt alo nto three dtnct component EL

4 VW EL 2 ( )( ) h (9) where VW 2( W / h) and EL 2( EL / h) are the well-known van der Waal and electrotatc component. Indeed, at low urface potental the electrotatc component of the urface tenon equal to, where the recprocal Debye length, 2 EL 0 tanh( h/ 2)/ 2 and the correpondng electrotatc djonng preure acqure EL 0 / 2coh ( h/ 2) t clacal form. 30 Let u conder now the lat adorpton component of the djonng preure n Eq. (9). To calculate t thckne dependence of adorpton one can employ the Maxwell relaton (8). Introducng the followng defnton hx X( h) X( ) for a dfference between the value of a property X of the equlbrum flm wth thckne h and nfnty, repectvely, one can wrte p. Note, that changng the flm thckne only t tangental preure component h L change, whle the normal one p p ( h ) reman contant. Thu, the Maxwell relaton (8) can be conecutvely modfed to G L pl 2( ) ( ) h h( ) h hc h (10) Knowng the adorpton otherm c ( ) at contant urface potental one able to ntegrate th equaton to obtan the thckne dependence of adorpton. If the change of the concentraton and adorpton, repectvely, are mall one can employ the followng lnear relatonhp h a hc, where a ( / c ) h the adorpton length on a ngle flat lqud/ga nterface. The latter, repreentng the thckne of the adorpton layer, depend on the adorpton equlbrum contant and. Solvng now the lnearzed dfferental equaton (10) yeld h h 0 exp( ) (11) 2a where { 0 } the dfference between the adorpton n a urfactant blayer ( h 0 ) and on a ngle flat urface ( h ). Subttutng now th expreon nto the defnton of the adorpton djonng preure from Eq. (9) lead to h 0 ( ) exp( ) (12) a 2a

5 Note that dependng on the gn of { 0 }, the adorpton djonng preure can be ether potve or negatve. It could be alo zero f no change n the adorpton n a blayer and on a ngle flat urface take place. Th probably the mot wdepread cae, whch explan why the adorpton djonng preure tll not well tuded. The preent thermodynamc theory cannot gve any value of { 0 } but jut aumng them decrbe the thckne dependence of ther effect. The dranage of thn lqud flm depend ubtantally on the moblty of flm urface. 25 Our preent etmate how, however, that wthn the pecfed SDS concentraton range the Marangon effect alway trong enough to block the tangental flow on the flm nterface. Hence, the dranage velocty can be well approxmated by the clacal Stefan-Reynold equaton dh dt 3 2 h ( p ) 2 3R (13) where p the capllary preure, R the flm radu, and the lqud vcoty. The djonng preure n Eq. (13) of crucal mportance for the modelng of the dranage. How t wa hown above, a uperpoton on the van der Waal, electrotatc and adorpton component. To determne the effect of the adorpton djonng preure correctly relable expreon for the DLVO component are requred. The van der Waal djonng preure between the flm urface can be etmated from the expreon 3 /6. Snce VW A h the flm thckne h alway larger than 150 nm and the Hamaker contant about 21 A 2 10 J, the van der Waal djonng preure neglgble for the preent ytem. At contant urface potental, the electrotatc djonng preure, calculated by the exact numercal oluton of the non-lnear Poon-Boltzmann equaton, em-analytcally decrbed a tanh ( / 4)[(1 coh ) nh ( / 4)exp( )] (14) EL RTc y h f y f h where f 2coh(0.332 y 0.779) for y 7, y F / RT a dmenonle urface potental on a ngle flat ar/oluton nterface. In the cae of a ngle urfactant Eq. (12) reduce to 0 exp( h/ 2 a)/ a (15) where the dfference of the adorpton component of the urface tenon on the blayer and on a ngle flat nterface. Here a the urfactant adorpton length. Snce could be ether potve or negatve dependng on the nteracton between the two monolayer

6 Flm thckne [nm] Flm thckne [nm] of the blayer, could be alo repulve or attractve, repectvely. In order to compare the above theory wth the expermental data, Eq. (13) wa numercally ntegrated ung a fourth-tep Runge-Kutta method. A macro wa wrtten n the VBA (Vual Bac for Applcaton) programmng language avalable from Mcrooft Excel. The meaured value for the zeta potental were adopted for the urface potental n EL.The adorpton length a are determned by the model of Kralchevky et al. 32 The computed flm thckne v. tme wa compared wth the expermental one for each of the SDS concentraton n order to obtan the bet ft of the free parameter (ee Fg. 1) M SDS Tme [] M SDS Tme []

7 Equlbrum flm thckne [nm] Flm thckne [nm] M SDS Tme [] Concentraton [M] Fgure 1. Example of expermental data and theoretcal ft of the flm thckne v. tme for ome SDS concentraton. The lat plot preent the equlbrum flm thckne v. SDS concentraton. Thee data along wth the capllary preure are preented n Table 1. The juxtapoton between the expermental and theoretcal flm thckne v. tme preented n the complementary materal attached to the paper. Table 1 how that the adorpton djonng preure potve nce 0 0, whch correpond to addtonal (non-dlvo) repulon between the flm urface. Th repulon orgnate from the overlap of the adorpton layer of the two flm urface. The latter ndcate that the adorpton component of the flm urface tenon ncreae wth the decreae of the flm thckne. Hence, the urfactant adorpton for th partcular

8 cae (SDS) dmnhe durng the flm thnnng. In general, the adorpton djonng preure hould dappear at zero SDS concentraton. A expected ncreae wth ncreang urfactant concentraton. Snce the adorpton length a reduce wth ncreang of c, the adorpton djonng preure become horter ranged and tronger. Table 1. Capllary preure p, zeta potental, adorpton length a and fttng parameter v. concentraton of SDS c [µm] p [Pa] [mv] a [nm] [µn/m] EL Djonng preure (Pa) p VW Flm thckne (nm) Fgure 2. Electrotatc EL, van der Waal VW, adorpton and total djonng preure v. the flm thckne for 10 µm SDS oluton. A general queton here how large the contrbuton of the adorpton djonng preure nto the total djonng preure. Another queton f th non-dlvo force long or hort

9 ranged. One can fnd anwer of thee queton n Fg. 2, whch how the electrotatc van der Waal EL, VW, adorpton and total djonng preure v. the flm thckne. The traght dahed lne n the fgure repreent the uckng capllary preure p. A een the adorpton nteracton between the flm urface decayng upon the flm thckne much weaker than the electrotatc and van der Waal nteracton. The contrbuton of the adorpton nteracton to the total nteracton between the flm urface gnfcantly mall. However, above a gven flm thckne (ca. 300 nm for the partcular cae of 10 M SDS) the adorpton nteracton preval over the electrotatc one. Depte beng long-ranged, the adorpton djonng preure cannot become equal to the capllary preure at any flm thckne due to t mall value. Th mean that n abence of electrotatc djonng preure an equlbrum flm cannot be formed and the flm wll thn untl rupture. The abence of adorpton nteracton, however, wll reflect n a gnfcantly maller equlbrum thckne of the flm (95 nm ntead of 160 nm) and fater flm dranage. The preent paper prove the extence of adorpton non-dlvo djonng preure between the foam flm urface. It orgnate by the overlap between the adorpton layer and can be attractve, repulve or vanhng. The adorpton djonng preure related to the properte of the adorpton layer. It part of the hydrophobc nteracton between the flm urface. 17 If the urfactant adorpton dmnhe upon the decreae of the flm thckne the adorpton nteracton repulve and vce vera. We menton here a well that uch flm develop treamng potental upon ther dranage. 33 Th theory valdated by experment on knetc of thnnng of foam flm from SDS wthn the concentraton range µm. Ft upon the parameter for each one of the concentraton performed. A expected ncreae wth ncreae on the urfactant concentraton. Thu defned the adorpton nteracton doe not dffer from th one defned by Tekov and Schulze 17 and Wang and Yoon. 12,13 A more detaled tudy for the effect of the adorpton otherm on the adorpton component of the djonng preure can be found n Ref Karakahev, S.I.; Manev, E.D.; Nguyen, A.V., Collod Surf. A 2008, 319, Exerowa, D.; Kolarov, T.; Khrtov, K., Collod Surf. 1987, 22, Buchavzov, N.; Stubenrauch, C., Langmur 2007, 23, Mhra, N.C.; Muruganathan, R.M.; Müller, H.-J.; Krutev, R., Collod Surf. A 2005, 256, Bowen, W.R.; Doneva, T.A.; Stoton, J.A.G., Collod Surf. A 2002, 201, Krutev, R.; Müller, H. J., Langmur 1999, 15, Iraelachvl, J.N.; Pahley, R.M.; Perez, E.; Tandon, R.K., Collod Surf. 1981, 2, Iraelachvl, J.; Pahley, R., Nature (London) 1982, 300, Rabnovch, Y.I.; Derjagun, B.V., Collod Surf. 1988, 30, Fa, K.; Nguyen, A.V.; Mller, J.D., J. Phy. Chem. B 2005, 109, 13112

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