The Four Kinetic Regimes of Adsorption from Micellar Surfactant Solutions

Transcription

1 The Four Kinetic Regie of Adorption fro Micellar Surfactant Solution P.A. Kralchevky, K.. anov, N.. enkov, N.C. Chritov, K.P. Ananthapadanabhan,* and Alex Lip* Laboratory of Cheical Phyic and Engineering, Faculty of Cheitry, Univerity of Sofia, Bulgaria, and *Unilever R&, Trubull, CT, USA Plenary Lecture at the 59 th iviional Meeting on Colloid and Interface Cheitry, The Cheical Society of Japan Hokkaido Univerity, Sapporo, Japan, Septeber 3-5, 006 Effect of icelle diffuion and diaebly on the dynaic urface tenion

2 ( The interfacial expanion give rie to urfactant adorption and to decreae in the onoer concentration near the interface; ( Thi lead to icelle decopoition and to diffuion of icelle. {Fat icellar proce} {Exchange of onoer between the icelle} (Rate contant k {Slow icellar proce} {ecopoition of a critical-ize icelle to onoer}. (Rate contant k S

3 General Set of Equation (Anianon & Wall, 974 Multi-Step Micellization: k + A + A A ( k,3, 4,... d c dt d c dt + I J 3 J + J J + (,3,4,... I iffuion and Reaction Fluxe: I J c (,,3,... k + c c k c (,3,4,... The proce i theoretically decribed by a yte containing ten of kinetic uation, which i inconvenient for application. For thi reaon, one of the baic proble of icellar kinetic i how to iplify the general et of uation without looing the aduacy and correctne of the theoretical decription.

4 Introduction of a Model (Gauian Micelle Size itribution c C ( exp[ π Anianon & Wall, 974 ] ( n r C C n r c n r c C n r ( c (i (ii (iii Region of the onoer and oligoer, Ω o ( n o Region of the rare aggregate, Ω r (n o < < n r Region of the abundant icelle, Ω ( n r New: ( We do not aue cont. ( We do not ue of the quai-uilibriu approxiation: local cheical uilibriu between icelle and onoer. (3 The derived general uation are nonlinear: applicable to both large and all perturbation.

5 Reduction of the Proble to 4 Equation for 4 Unknown Function Monoer concentr.; Total icelle conc.; Mean aggreg. nuber; Polydiperity c (r,t, C (r,t, (r,t, (r,t d c dt + I n r J J,0 Nonlinear expreion for the fluxe J, J,0, and J, are derived. dc dt d d t ( C + I,0 + I J, n r J + J,0 J J n, J ( r, i J i > n r i 0, d d t [( + C ] + I, n r J J,0 + J, i I, i I ( i n r 0,,

6 Relaxation of a Spatially Unifor Perturbation (C jup; T jup; P jup: bulk relaxation ethod Our purpoe i to ee what are the prediction of the odel for thi type of perturbation. ienionle perturb. in: Total ic. conc.; Mean aggreg. nuber; Polydiperity Linearization of the proble: Syte of three linear uation for ξ c, ξ, and ξ ξ c C C,p, A hoogeneou yte ha a nontrivial olution only if it deterinant i ual to zero characteritic uation I, I, I 3 invariant of the atrix (a ij ; three eigenvalue three relaxation tie: τ c, τ, τ. ; ξ j c,, ( a 3 p ij ; ξ p λδ ξ 0, i c,, λ I λ ij + j I λ I 3 0 λ j ; j c,, τ k t j j

7 The Three Micellar Characteritic Relaxation Tie λ j ; j c,, τ k t j j C tot β CMC CMC ienionle icelle concentration t c k S β β + + β / t c i the characteritic tie of the low proce, that i the relaxation of the total icellar concentration C ; k S rate contant of the low proce (Anianon-Wall + k t β t i the firt characteritic tie of the fat proce, related to the relaxation of the ean icellar aggregation nuber, ; k rate contant of the fat proce (Anianon-Wall t k t i the econd characteritic tie of the fat proce, related to the relaxation of the icellar polydiperity, ; (new! k rate contant of the fat proce. The expreion for t c and t by Anianon & Wall are confired!

8 Nuerical Reult for Typical Paraeter Value: 7 60; 5; ks / k 0 ienionle relax. tie Exact Low icelle concentration: β Approxiate expreion τ c τ τ High icelle concentration: β 00 τ c τ τ ienionle icelle concentration: C tot β CMC CMC Low icelleconcentration τ > : c > τ τ High icelleconcentration : τ > τ > c τ τ c relaxation tie of icelle concentration (of the low proce, Anianon & Wall τ relaxation tie of ean aggreg. nuber (of the fat proce, Anianon & Wall τ relaxation tie of icelle polydiperity (new effect, predicted by preent theory

9 Ayptotic Expreion: The three baic phyical paraeter, C, and, are perturbed. ( The Relaxation of a Perturbation in Micelle Concentration, C, i Governed only by the Slow Micellar Tie, t c ξc ( t Ac exp( t / tc ( tc >> t, t ( A c aplitude of the perturbation in C Hence, if the relaxation of icelle concentration i eaured, only the Slow Micellar Tie, t c, could be deterined. Note, however, that a perturbation in C perturb alo and!

10 The Relaxation of a Perturbation in Micelle Mean Aggregation Nuber i Governed by both the Slow and Fat Tie, t c and t ξ ( t A c τ β exp( t / t c + ( A + A c τ β exp( t / t ( A c and A aplitude of the perturbation in C and Hence, if the relaxation of icelle ean aggregation nuber i eaured, then both the low and fat icellar tie, t c and t, could be deterined. Note that a perturbation in perturb, but doe not affect C

11 The Relaxation of a Perturbation in Micelle Polydiperity i Governed by the two Fat Relaxation Tie, t and t Hence, if the relaxation of icelle polydiperity i eaured, then the two fat icellar tie, t and t, could be deterined. Note that a perturbation in polydiperity doe not affect C and. ] / exp( / [exp( ( / exp( ( β τ ξ t t t t t t t A A t t A t c + +

12 Suary of Part : Micellar Relaxation Procee in the Bulk The theoretical analyi iplie: (A The relaxation of the three baic paraeter, the icelle concentration, C, the ean aggregation nuber,, and the polydiperity,, are characterized by three ditinct relaxation tie: t c, t, and t. (B The firt two of the, t c and t, coincide with the conventional low and fat icellar relaxation tie. (C The third relaxation tie, t, i cloe to t for low icelle concentration, but at high icelle concentration we have t c > t > t..

13 ( The relaxation of C i affected by t c alone. (E The relaxation of i affected by both t and t c. (F The the relaxation of i affected by t and t. Siple, but accurate analytical expreion are available: ( For calculation of the three relaxation tie; ( For decribing the evolution of a icellar yte. [K.. anov, P.A. Kralchevky, et al., Adv. Colloid Interface Sci. 9 (006-6] Next tep: Invetigation of the proble about the kinetic of adorption fro icellar olution, and the repective dynaic urface tenion (Part.

14 Part : Theoretical odeling of adorption fro icellar olution at quiecent and expanding urface Main quetion to be anwered: Why in different cae different kinetic regie are oberved? (a diffuion liited kinetic: [ Δ t / ]; (b reaction liited kinetic: [ Δ exp( t /τ ]. Which of the two very different theoretical expreion for the effective diffuivity of a icellar olution i correct? (a by J. Lucaen (975: eff ( + β ( + β / (b by Paul Joo (988: eff ( + β ( + β /

15 Paraeter of the Adorption Proce ζ dienionle ditance; τ dienionle tie; diffuivity of the onoer; k rate contant of the low icellar proce; k rate contant of the fat proce. a S a a a ; ; ; k h K k h K t h z h τ ζ (adorption length ( a c h Γ

16 ienionle Perturbation c C ( exp[ π ] ( n r ξ Γ h p a (0 c,p ; ξ c h β Γ a p C (0,p ; ξ h a c Γ, p (0 p ; ξ h a c Γ, p p (0 ξ dienionle perturbation in the concentration of onoer, c ; ξ c dienionle perturbation in the icelle concentration, C ; ξ dienionle perturbation in the icelle ean aggregation nuber, ; ξ dienionle perturbation in the icelle polydiperity,.

17 General Syte of Kinetic Equation (fro Part S K S K w ϕ β ϕ ζ ξ τ ξ ( c c K B ϕ β ζ ξ τ ξ + K w K B ϕ ϕ β ζ ξ τ ξ + ξ ϕ ϕ β ζ ξ τ ξ ( K K w K B + (urfactant onoer (concentration of icelle (icelle ean aggregation nuber (polydiperity ϕ dienionle reaction flux of the low relaxation proce; ϕ dienionle reaction flux of the fat relaxation proce. CMC ( tot CMC / C β B

18 Reaction fluxe fro the low and fat relaxation procee ϕ ξ ξ For ξ ξ, we obtain ϕ 0 (criterion for uilibriu with repect to the fat icellization proce ϕ ( w ξ ξ + c wξ For ξ ξ c ξ, we obtain ϕ 0 (criterion for uilibriu with repect to the low icellization proce Method of olution of the general yte of linear partial differential uation: Laplace tranfor, olving the uation, and nuerical revere Laplace tranfor

19 Nuerical Reult: Typical Relaxation Curve (Four kinetic regie of adorption: AB, BC, C, and E γ ( t γ γ (0 γ ξ,0 ( τ ξ,0 decribe the relaxation of the urface tenion γ (t. Four different relaxation regie: AB, BC, C, E (B ξ ξ, then ϕ 0 uilibrated fat icellar proce; ( ξ ξ c ξ ϕ ϕ 0 uilibrated fat and low icellar procee.

20 Analytical Expreion for the Relaxation in ifferent Regie τ θ F θ τ C θ c θ c dienionle relaxation tie of the low proce; θ and θ d.le relaxation tie of the fat proce. Two exponential regie (AB and C with relaxation tie τ F and τ C ; Two invere-quare-root regie (BC & E with relaxation tie τ BC and τ E.

21 ( Kinetic Regie AB The fat icellar proce govern the adorption kinetic [exp(t/τ F ] τ / F, F ( + β K θ θ τ F 4 ξ,0 F exp( F F τ + π 0 τ exp[ ( + ττ~ τ F ] ( ~ τ ~ τ + / τ F + ~ τ d ~ τ τ / ξ,0 ( + τ +... (hort π F F ξ,0 exp( τ +... (long F tie ayptotic tie ayptotic τ F F The regie AB(exp wa oberved by P. Joo for Triton X-00, inclined plate ethod. (epending on the urfactant and experient. ethod, different regie are oberved!

22 ( Kinetic Regie BC ξ,0 ξ,0 iffuion control the fat icellar proce i uilibrated, wherea the effect of the low proce i negligible [t / ] τ BC / ξ,0 ( +... π τ (invere-quare-root tie dependence τ BC BC ( + β ( + β (relaxation tie τ BC, & effective diffuivity BC The regie BC wa oberved by u for SS with the axiu bubble preure ethod (MBPM, and by Makievki et al. for Triton X-00, MBPM again. For /, the expreion for BC reduce to that propoed by Joo (988.

23 (3 Kinetic Regie C The low icellar proce govern the adorption kinetic [exp(t/τ S ] γ ( t γ γ (0 γ ξ,0 ( τ ξ,0 τ BC ( π τ / τ exp( τ C (exponential relaxation τ C θ c β 3 K (the relaxation tie, τ C, coincide with the characteritic tie of the low icellar proce

24 (4 Kinetic Regie E iffuion control both the fat and low icellar procee are uilibrated [t / ] τ E / ξ,0 ( +... (invere-quare-root-of-tie dependence; Lucaen 975 π τ τ E + E + ( + β ( + β (Expreion for the Relaxation Tie & iffuivity The Lucaen uation wa unuccefully tried by Joo et al. to fit data for Brij 58 (trip ethod. It turn out that the data by Joo et al. (988 correpond to the regie BC, which ha not been identified at that tie.

25 Cobined Expreion for the whole BCE region Analyzing the baic yte of uation we arrived at the following cobined forula for the whole region BCE: ξ,0 τ E ( π τ / τ BC + ( π τ / τ exp( τ C τ E + + ( + β ( + β τ BC ( + β ( + β 00 >> / τ << τ E BC (eay to ditinguih regie BC fro E [etail in: K.. anov, P.A. Kralchevky, et al., Adv. Colloid Interface Sci, 9 ( ]

26 ifference between the exponential regie AB and C /τ F increae with icelle concentration; /τ C decreae with icelle concentration Exponential regie AB: ξ,0 τ exp( τ + τ F Exponential regie C: τ BC / τ ξ,0 ( exp( π τ τ F C

27 Surfactant v. Method in Relation to the Relaxation Regie Fat urfactant {urfactant that adorb quickly}: Fat ethod {ethod that eaure early urface age}: Regie AB: Can be detected for low urfactant by fat ethod. Exaple: Triton X-00 by inclined plate ethod. Regie C and E: ifficult for detection becaue ξ,0 ha becoe very all for thee regie; in principle, thee regie could be detected for very fat urfactant by very low and enitive ethod. Regie BC: Can be detected for fat urfactant by low ethod. Exaple: SS by MBPM.

28 Rudientary Kinetic iagra at Low Micelle Concentration (β and/or at Saller ifference between the Rate of the Fat and Slow Micellization Procee (aller K /K The three icellar relaxation tie, θ c, θ and θ, are cloe to each other. Point A, B, and C coincide. τ / ξ,0 ( + τ +... π (hort tie ayptotic (the ae a in regie AB The effect of the fat relaxation proce diappear at low urfactant concentration (β ξ,0 τ exp( τ + τ C The long-tie ayptotic i the ae a in regie AB but τ F τ C C

29 Method with Stationary Interfacial Expanion: Micellar Kinetic d A dα ( t α& ( t A d t d t cont. The kinetic regie are: (AB θ /(K AB + θ / kinetic regie governed by the fat icellization proce; (BC θ / diffuion-liited regie at uilibrated fat proce but negligible low proce; (C θ /(K C + θ / kinetic regie governed by the low icellization proce; θ a h & α θ dienionle expanion rate (E θ / diffuion-liited regie at uilibrated both the fat and low icellization procee. [etail in: K.. anov, P.A. Kralchevky, et al., Colloid & Surface A, 8-83 ( ]

30 Coparion of Theory and Experient Exaple : ynaic urface tenion of Brij 58 eaured by the trip ethod (Paul Joo Theory: ΓkT π & α / ( CMC BC CMC ol/c 3 ifferent curve correpond to different icelle concentration, β Γ ol/c (at CMC Fro the lope of the plot α& / v. at fixed β, one deterine the apparent diffuivity BC (β.

31 Coparion of Theory and Experient (Continued Exaple : ynaic urface tenion of Brij 58 eaured by the trip ethod (Paul Joo 00 Brij 58, Strip ethod u.06, B BC + uβ ( + uβb ( BC / B / 0.43 (reaonable value β (C tot CMC/CMC Fro the fit of the data, one deterine: u For Brij 58, 70 ; hence the polydiperity of the icelle i 8.6 The obtained reaonable value confir that the kinetic regie i BC /.06

32 The Overflowing Cylinder Method Another ethod with tationary interfacial expanion d A α& ( t cont. A d t Adorption (rather than urface tenion i detected by ellipoetry or neutron reflection The Overflowing Cylinder Method (OFC C.. Bain et al. Languir 004, 0,

33 Exaple : ynaic urface tenion of C 4 TAB by the Overflowing Cylinder Method (Colin Bain et al.; the adorption Γ i directly eaured by ellipoetry The data point are not for the ae β! Theory: Y Γ Γ Y 0 (i.e. 0, give the uilibriu adorption at CMC, Γ. / τ dif ( Γ Y { π & α /[( + uβ ( + uβb ]} α& / Adorption, Γ (μol/ Γ C 4 TAB + 00 M NaBr Overflowing cylinder u.; B 0.3 The bet fit Γ v. Y correpond to B / 0.3 and u / Y ( / 80; fro u. we deterine that the polydiperity of the C 4 TAB icelle i 9.8. The obtained value of Γ. B and are reaonable. Thi confir that the kinetic regie i BC. [τ dif h a / i the characteritic diffuion tie]

34 Part 3: Application of the Maxiu Bubble Preure Method Proble: ifferent tenioeter - different reult for the dynaic urface tenion. Thi i difference i deontrated with our data for two apparatue. The data are plotted a ST v. t / : ynaic urface tenion, N/ M SS, 8 M NaCl t / ( / Kru BP Setup - Horozov Explanation: ifferent tie-dependence, A(t, of the bubble urface area for different apparatue.

35 Solution of the Proble Suggeted by the Experient The experient indicate that A(t d i independent of:. Bubbling period, t age. Surfactant concentration 3. Surfactant type (General validity? The theory indicate that in ot cae γ(t depend on a contant paraeter, λ integral of A(t, rather than on the function A(t. A(t (the apparatu function can be deterined only by cineatography; λ (the apparatu contant can be deterined alo by MBPM (uch eaier! Below we check whether λ i independent of t age, urfactant type and concentration.

36 Expanding Surface v. Iobile Surface γ γ,0λ γ γ + γ / + γ / + ( t ( t ( t age age u γ,0 / γ kt Γ λ γ, 0 λ, t / u tage / ( π c λ ( τ τ t d τ 0 0 A d A0 (ˆ t dˆ t / d, d dt d A( t [ A d 0 ]dt τ τ ( t d d λ ( The whole effect of the interfacial expanion i incorporated in λ ; ( t u (univeral urface age, i the age of an (initially clean iobile urface with the ae γ a that regitered by the MBPM tenioeter. c bulk urfactant concentration; Γ uilibriu adorption; γ uilibriu urface tenion; urfactant diffuivity; γ,0 the value of γ for an iobile interface.

37 Exaple : Coparion of data obtained by MBPM (expanding bubble with data for iobile bubble (IB for SS + 00 M NaCl. For MBPM λ i deterined by integration of the experiental A(t curve (for our apparatu; For IB we have λ (no expanion t u t age /λ i ued to plot the MBPM data in (b (λ 37

38 70.5 M NaCG + 00 M NaCl Exaple : Surface tenion, γ (N/ MBPM Wilhely plate Coparion of data obtained by MBPM (expanding bubble with data fro Wilhely plate ethod for Na N-Cocoylglycinate Noinal urface age, t age / ( / For Wilhely plate (iobile urface: t u t age (λ For MBPM (expanding urface: t u t age /λ (λ The excellent fit with the theoretical dependence evidence diffuion-liited adorption kinetic. Surface tenion, γ (N/ 70.5 M NaCG + 00 M NaCl 60 MBPM 50 γ γ γ γ,0 / γ,0 + tu 4.43 ± 0.0 N/ Wilhely plate / / Univeral urface age, t u ( + a

39 Two way to deterine the apparatu contant, λ: ( By integration of the experiental A(t curve; ( By MBPM experient, λ γ / γ,0. γ,0 kt (π Γ / c (Copare the value of λ obtained in the two way! Procedure: ( γ (t age curve are obtained by MBPM and fitted with the dependence: γ γ + a + (t age / Thu γ i deterined. ( Next, γ,0 i calculated fro fit of uilibriu urface tenion iother. γ (3 λ γ / γ,0

40 C SS (M Γ (μol/ γ,0 (N.. / γ (N.. / λ γ / γ,0 SS + 0 M NaCl SS + 00 M NaCl Theory: λ Average: λ 6.07 ± 0.0

41 C TAB (M Γ (μol/ γ,0 (N.. / γ (N.. / λ γ / γ,0 TAB + 5 M NaBr TAB + 00 M NaBr Theory: λ Average: λ 6.07 ± 0.0

42 Concluion: The reult confir the concept about the apparatu contant: ( λ i the ae for all concentration of a given urfactant and electrolyte; ( λ i the ae for SS and TAB; (3 λ i calculated fro γ deterined of the data fit for 0 < t age < 40 t u t age /λ λ ( Hence, in ter of t u the MBP ethod i uch (37 tie fater. t u t age /λ, account for the urface expanion, and for thi reaon t u give the phyically correct urface age. etail in: Chritov et al., Languir (

43 Plot of the dienionle effective diffuivity of icellar olution, eff /, v. β Maxiu Bubble Preure Method (MBPM eff γ,cmc γ diffuivity of urfactant onoer. The data for γ(t are fitted well with the expreion for diffuion-liited adorption: γ γ + a + (t γ age / eff BC ( + β ( + β Hence, the kinetic regie i either BC or E. The data coplie with BC!

44 Plot of Plot of v. β calculated fro the data for eff / (MBPM Rudientary kinetic regie icelle ean aggregation nuber; icelle polydiperity Kinetic regie BC Reaonable paraeter value the kinetic regie i BC Value / < are reaonable. For exaple, for β 0 and 70 we obtain: 7. (SS 9.5 (TAB

45 Suary and Concluion The theory indicate the preence of four different kinetic regie of adorption fro icellar urfactant olution: ( Regie AB: the fat icellar proce govern the adorption kinetic [exp(t/τ F ] ( Regie BC: diffuion control the fat icellar proce i uilibrated, wherea the effect of the low proce i negligible [t / ]. (3 Regie C: the low icellar proce govern the adorption kinetic [exp(t/τ S ] (4 Regie E: diffuion control both the fat and low icellar procee are uilibrated [t / ]. (5 MBPM: The deterination of the apparatu contant, λ, for a given tenioeter allow one to characterize a given urfactant olution with a univeral dynaic urface tenion curve, γ(t.

46 The reult are publihed in the following paper:. K.. anov, P.A. Kralchevky, N.. enkov, K.P. Ananthapadanabhan, and A. Lip, Ma Tranport in Micellar Surfactant Solution:. Relaxation of Micelle Concentration, Aggregation Nuber and Polydiperity, Adv. Colloid Interface Sci. 9 ( K.. anov, P.A. Kralchevky, N.. enkov, K.P. Ananthapadanabhan, and A. Lip, Ma Tranport in Micellar Surfactant Solution:. Theoretical Modeling of Adorption at a Quiecent Interface, Adv. Colloid Interface Sci. 9 ( K.. anov, P.A. Kralchevky, K.P. Ananthapadanabhan, and A. Lip, Micellar Surfactant Solution: ynaic of Adorption at Fluid Interface Subjected to Stationary Expanion, Colloid & Surface A, 8-83 ( N.C. Chritov, K.. anov, P.A. Kralchevky, K.P. Ananthapadanabhan, and A. Lip, The Maxiu Bubble Preure Method: Univeral Surface Age and Tranport Mechani in Surfactant Solution, Languir ( K.. anov, P.A. Kralchevky, K.P. Ananthapadanabhan, and A. Lip, Influence of Electrolyte on the ynaic Surface Tenion of Ionic Surfactant Solution: Expanding and Iobile Interface, J. Colloid Interface Sci. (006 in pre.