Kinetic Theory of Molecular Mechanism of Micellar ... - CSJ Journals

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Sep 29, 2012 - 2Faculty of Chemistry, St Petersburg State University, St Petersburg 198504, ..... aggregation numbers to balance equations given below:.
doi:10.1246/cl.2012.1081 Published on the web September 29, 2012

1081

Kinetic Theory of Molecular Mechanism of Micellar Relaxation Alexander K. Shchekin,*1 Anatoly I. Rusanov,2 and Fedor M. Kuni1 Faculty of Physics, St Petersburg State University, St Petersburg 198504, Russia 2 Faculty of Chemistry, St Petersburg State University, St Petersburg 198504, Russia 1

(Received May 14, 2012; CL-120507; E-mail: [email protected]) Analytical methods for finding characteristic times and overall time behavior of moments of micelle distribution during fast and slow micellar relaxation in the case of the coexistence of spherical and cylindrical micelles are considered based on Becker­Döring kinetic equation. The linearized and nonlinear forms of this equation are studied, and its solutions are derived.

Studies on relaxation phenomena in micellar solutions of surfactants have always been one of the major sources of information on the structure and properties of micelles.1,2 In the following section, we will be dealing with an analytical description of relaxation in solutions of nonionic surfactant above the second critical micelle concentration, cmc2. In this case, the surfactant is accumulated mostly in spherical and cylindrical micelles, and the fraction of cylindrical micelles rapidly increases as the total surfactant concentration grows. Our aim here is to present a compact but, as much as possible, a complete and consistent theory of the molecular mechanism of micellar relaxation based on previous and recent results, with new improvements and generalizations extending the applicability of the theory. Molecular mechanism of micellar relaxation assumes that all transitions between small molecular aggregates, spherical and cylindrical micelles are realized as a chain of captures and emissions of single surfactant molecules. Evolution of concentration of aggregates (cn ), where n is the aggregation number, is governed during such mechanisms by the Becker­Döring kinetic equation1,3­5 given below: @cn =@t ¼ ðJn  Jn1 Þ;

n ¼ 2; 3; . . .

ð1Þ

where t is the time and Jn ¼ an c1 ðtÞcn ðtÞ  bnþ1 cnþ1 ðtÞ is the flux of aggregates along the aggregation number axis. In the definition of Jn, an c1 is the number of monomers attached to the aggregate fng per unit time in a solution with a monomer concentration c1 , and bnþ1 is the number of monomers detached from the aggregate fn þ 1g into the solution per unit time. Relaxation tends to an aggregate and JnðeqÞ ¼ 0, which gives the detachequilibrium with cn ¼ cðeqÞ n ðeqÞ ðeqÞ ment rate as bnþ1 ¼ an cðeqÞ c n =cnþ1 . 1 Aggregate equilibrium can be local or global. In any case, an equilibrium distribution of aggregates is a Boltzmann distribution of the form cðeqÞ / eWn , where Wn is the aggregation work (i.e., the n minimal work required for the formation of surfactant aggregate and is expressed in thermal units of kB T , with kB being the Boltzmann constant and T being the absolute temperature). For total surfactant concentrations above the cmc2, the aggregation work as a function of the aggregation number is shown in Figure 1. It has its first peak with a half-width nð1Þ (premicellar aggregates) and at n ¼ nð1Þ c c narrow well at n ¼ ns with a half-width ns (stable spherical micelles). For larger aggregation numbers, Wn has a second peak at ð2Þ n ¼ nð2Þ c with a half-width nc and minimum rolling in a wide slow linear increase at n  n0 , where cylindrical micelles with large aggregation number n and width n  n0 accumulate.5­10 After initial disturbance of a micellar solution, two relaxation processes with different time scales can be observed. First process

Chem. Lett. 2012, 41, 1081­1083

Figure 1. The aggregation work Wn as a function of aggregation number n above the cmc2.

develops in a relatively short time with local barrierless restructuring (via exchange by monomers) of the distribution of micelles to the most probable distribution at the current state of parameters of the system for a given number of micelles. It is called fast relaxation.1­5,9­11 Process associated with the barrier transitions from premicellar aggregates to spherical micelles and vice versa, or with the barrier transitions between spherical and cylindrical micelles usually requires more time and is called slow relaxation.1­4,12­14 As a result of fast relaxation, two separate local equilibrium concentrations of spherical and cylindrical micelles are established5,8,14 as given below: _

_

_

_ cn

_

_

_

c SM

_

c n ¼ c s eðW n W s Þ 

_

³1=2 n s

_

¼ c 0 eðW n W 0 Þ 

_ c CM

_ 2

eðn n s Þ

_

= n s 2

;

_

_

jn  n s j  n s ð2Þ

nn _ 0 n  n0 ;

n  n0 ð3Þ n   n0 where, the symbol _ denotes quantities measured at local _ _ equilibriums at the end of fast relaxation and c SM and c CM are the total concentrations of the spherical and cylindrical micelles, respectively. Local concentrations in eqs 2 and 3 are separate in the _ _ sense that the total concentrations c SM and c CM are determined by the initial state of the system. They stay fixed during the fast relaxation and are not parts of the one-piece equilibrium distribution of aggregates. To describe fast relaxation in an isolated system, it is _ _ convenient to pass to relative deviation ²n ¼ ðcn  c n Þ= c n from the local equilibrium distribution. Equation 1 can be rewritten for ²n _ _ in the differential form at jn  n s j  n s as   _ @½ð1 þ ²n ðtÞÞ c n  _ @²n ðtÞ _ @ _ @²n ðtÞ _ cn ¼ a_n s c 1 cn  a_n s c 1 ²1 ðtÞ @t @n @n @n and at n  n0 as _ cn

_

e

ð4Þ

_   a_n  c 1 @ @²n ðtÞ _ @² ðtÞ ¼_ ðn  n0 Þ c n n @t @n n   n0 @n

© 2012 The Chemical Society of Japan

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1082 _

a_ c 1 @½ð1 þ ²n ðtÞÞn_ c n  ²1 ðtÞ _ n  ð5Þ @n n   n0 It has been recognized here that an ’ a_n s is for spherical micelles _ _ _ at jn  n s j  n s and an ’ a_n  ðn  n0 Þ=ðn   n0 Þ is for cylindrical micelles at n  n0 . The condition of isolation, c1 þ P n¼2 ncn ¼ c (c is the total surfactant concentration), can be rearranged in the continuous form as Z1 _ _ c 1 ²1 ðtÞ ¼  dnn c n ²n ðtÞ ð6Þ 2

In view of eq 6, eqs 4 and 5 are coupled nonlinear inhomogeneous differential equations. For small ²n ðtÞ  1, the nonlinear last terms in eqs 4 and 5 can be dropped out. In this case, the solutions _ _ of eqs 4 and 5 can be represented as Hermite, Hi ððn  n s Þ=n s Þ _ (i = 1, 2, +) and Laguerre, Li ððn  n0 Þ=ðn   n0 ÞÞ (i = 1, 2, +) series of orthogonal polynomials. The choice of the polynomials is provided by eqs 2 and 3, which determine the weight and generating functions just for these polynomials. The solutions can be written as ! 8 1 _ X > n  ns _ _ > > (jn  n s j  n s ) mi ðtÞHi > _ < n s i¼1 ²n ðtÞ ¼ ð7Þ   1 > X > n  n0 > > qi ðtÞLi _ (n  n0 ) : n   n0 i¼1 _

c CM

_

c SM _ ð8Þ _ n s m1 ðtÞ c1 c1 where functions mi ðtÞ and qi ðtÞ are expressed through the moments of the linearized distribution of micelles in aggregation numbers, m0 ðtÞ ¼ 0 and q0 ðtÞ ¼ 0. Substituting eqs 7 and 8 in eqs 4 and 5, respectively, and using the orthogonality of the polynomials leads to the following expressions: ²1 ðtÞ ¼

_

_

ðn   n0 Þq1 ðtÞ 

mi ðtÞ ¼ mi ð0Þet=¸ si ; ¸ si ¼

_ ðn s Þ2 _ 2a_n s c 1

qi ðtÞ ¼ qi ð0Þet=¸ci ;

1 ; i

¸ ci ¼

_ ðn 

ði ¼ 2; 3; . . .Þ

 n0 Þ 1 ; _ a c1 i 2

ð9Þ

ði ¼ 2; 3; . . .Þ ð10Þ

_

n

m1 ðtÞ ¼ A1 et=¸ s1 þ A2 et=¸c1 ;

¸ 1 s1 ¸ 1 c1

q1 ðtÞ ¼ B1 et=¸s1 þ B2 et=¸c1  1=2 1 1 ð¢11  ¢22 Þ2 þ ¢12 ¢21 ¼ ð¢11 þ ¢22 Þ þ 2 4  1=2 1 1 ð¢11  ¢22 Þ2 þ ¢12 ¢21 ¼ ð¢11 þ ¢22 Þ  2 4 ! _ _ _ 2a_n s c 1 c SM n s 2 ; ¢11 _ 1þ _ n s 2 2c1 _

¢12 a_n s ¢21 a_n 

_

c SM n s _

n   n0

;

ðn   n0 Þ2

_



_

c CM ðn   n0 Þ2 _

c1

! ð15Þ

Here, we have improved the previous results8,10 for the overall time behavior at fast relaxation and found a set of specific times of fast relaxation in systems with coexisting spherical and cylindrical micelles. These times are associated with different exponential contributions from various moments of the distribution function at the stage when the distribution function is close to the local equilibriums for given total concentrations of spherical and

Chem. Lett. 2012, 41, 1081­1083

_

_

a_ c 1 d MkCM ¼ k _ n  dt n   n0

n s

a_n  c 1

_

ð13Þ

ð14Þ

_

_

¢22

ð12Þ

_

c CM ðn   n0 Þ

_

ð11Þ

cylindrical micelles. Certainly, the largest time here is of special interest as a scale time of exponential fast relaxation. As follows from eqs 10 and 12­15, the hierarchy of kinetic times depends upon the total surfactant concentration and model of the aggregation work _ _ through such characteristics as n s and n  . Equations 7­15 allow one to consider the cases of only spherical or only cylindrical _ _ micelles by setting c CM ¼ 0 or c SM ¼ 0. The solutions of the linearized Becker­Döring kinetic equations do not allow us to predict the whole time behavior of fast relaxation for large initial disturbances in the system. We cannot even use the same approach of separating variables for nonlinear kinetic equations, and it means that the whole time behavior cannot be described by exponential functions of time. However, we can convert nonlinear kinetic eqs 4 and 5 for the aggregate distribution function into nonlinear equations for the moments of this distribution function and solve them. In the following section, we will extend the previous results.5 Let us define the moments of the distribution function for spherical (superscript SM) and cylindrical (superscript CM) micelles using the relations Z 1 _ dnnk c n ; ðk ¼ 0; 1; 2; . . .Þ ð16Þ MkSM _ _ _ c 1 jn n s j n s Z 1 _ MkSM _ dnnk c n ²n ðtÞ; ðk ¼ 0; 1; 2; . . .Þ ð17Þ _ _ c 1 jn n s j n s Z 1 _ CM Mk _ dnnk c n ; ðk ¼ 0; 1; 2; . . .Þ ð18Þ c 1 nn0 Z 1 _ MkCM _ dnnk c n ²n ðtÞ; ðk ¼ 0; 1; 2; . . .Þ ð19Þ c 1 nn0 Multiplying both parts of eqs 4 and 5 by nk, integrating respectively the result over the first potential well of the aggregation work and over the interval n  n0 , applying integration by parts twice, and using the equality ²1 ðtÞ ¼ M1SM  M1CM which follows from eq 6, we find  SM d _ _ 2Mk MkSM ¼ k a _n s c 1 _ 2 dt ðn s Þ ! _ 2n s SM CM SM Mk1   M  M _ 1 1 ðn s Þ2  SM SM  ðk  1ÞMk2 þ ðM1SM þ M1CM ÞMk1 ð20Þ





1

 þ M1CM þ M1SM

_ n   n0 CM CM ðMk  n0 Mk1 Þ CM CM kMk1 þ ðk  1Þn0 Mk2

CM þ ðM1CM þ M1SM ÞðMkCM  n0 Mk1 Þ

 ð21Þ

As observed from eqs 20 and 21, the higher moments can be expressed through the lower ones, and the problem is reduced to the successive finding of moments of increasing order. From a practical standpoint, the most important moments are that with k = 0 and 1. It follows from the constancy of the total concentrations of spherical and cylindrical micelles at fast relaxation, and from eqs 16, 18, 20, _ _ and 21 that M0SM ¼ 0, M0CM ¼ 0, M0SM ¼ c SM = c 1 , M0CM ¼ _ _ c CM = c 1 , " _ ! # _ d c SM 2 c SM _ _ SM SM CM _ M1 þ _ M1 M1 ¼ a n s c 1 _ þ _ dt c1 c1 ðn s Þ2

© 2012 The Chemical Society of Japan

ð22Þ

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1083 _

_

a_ c1 d M1CM ¼  _ n  dt n   n0



_

c CM

!

_

1 c CM þ_ M1CM þ _ M1SM n   n0 c1 c1  þ ðM1SM þ M1CM ÞM1CM ð23Þ _

Solutions of eqs 22 and 23 can be found analytically in the cases of only spherical or only cylindrical micelles. In other cases, eqs 22 and 23 can be easily solved numerically. The results show that there is a strong interplay between spherical and cylindrical micelles, even in the case when the total number of surfactant molecules aggregated in spherical micelles is small in comparison with that in cylindrical micelles. The mentioned difference in local equilibriums from the global equilibrium for oligomers, spherical, and cylindrical micelles for all aggregates leads to the direct and backward fluxes of aggregates over first and second potential peaks of the aggregation work.3,4,8 During slow relaxation, these fluxes can be considered as quasisteady. Using this, eq 1 can be converted by summation over the aggregation numbers to balance equations given below: dcSM =dt ¼ J10  J100  ðJ20  J200 Þ;

dcCM =dt ¼ J20  J200

c ¼ c1 þ ns cSM þ n cCM

ð24Þ ð25Þ

Where, quasi-steady direct fluxes (marked with a prime and subscript refer to the potential peak) and corresponding backward fluxes (indicated by two strokes) are defined8,14 as ð1Þ 2 expðWc Þ J10 ¼ anð1Þ c ; 1 ð1Þ c ³1=2 nc exp½ðWcð1Þ  Ws Þ J100 ¼ anð1Þ c1 cSM ð26Þ c ³nð1Þ c ns exp½ðWcð2Þ  Ws Þ c1 cSM ; J20 ¼ anð2Þ c ³nð2Þ c ns exp½ðWcð2Þ  W0 Þ c c ð27Þ J200 ¼ anð2Þ 1 CM c ³1=2 ðn  n0 Þnð2Þ c These nonlinear (in monomer concentration) equations describe the whole stage of slow relaxation at arbitrary initial deviation from equilibrium. With known model of the aggregation work, such as the droplet or quasi-droplet model3,15,16 that determines Wn as a function of the aggregation number and monomer concentration, all the quantities in eqs 25­27 can be defined and eq 24 can be integrated. Note that it can be done analytically for any model in the case of small deviations from the final equilibrium of the system with coexisting spherical and cylindrical micelles. Marking the quantities in the final equilibrium by upper tilda and introducing small deviations ¤c1 ðtÞ c1 ðtÞ  c~ 1 , ¤cSM ðtÞ cSM ðtÞ  c~ SM , and ¤cCM ðtÞ cCM ðtÞ  c~ CM and using thermodynamic equalities17 @Wn n1 ¼ ; c1 @c1

@ns ðns Þ2 ¼ ; @c1 2c1

@n ðn  n0 Þ2 ¼ @c1 c1

ð28Þ

one can reduce eqs 24 and 25 to the following coupled linear equations d¤cSM ¼ ¡11 ¤cSM  ¡12 ¤cCM ; dt d¤cCM ¼ ¡21 ¤cSM  ¡22 ¤cCM dt

ð29Þ

where ¡11

0 0 0 0 J~1 þ J~2 n~ s ð~ns J~1  ð~n  n~ s ÞJ~2 Þ þ c~ SM c~ 1 þ ~n2s c~ SM =2 þ ð~n  n0 Þ2 c~ CM

Chem. Lett. 2012, 41, 1081­1083

ð30Þ

_ 0 0 0 n  ð~ns J~1  ð~n  n~ s ÞJ~2 Þ J~ ¡12  _ 2 þ 2 c CM c~ 1 þ ~n2s c~ SM =2 þ ð~n  n0 Þ c~ CM 0 0 J~ n~ s ð~n  n~ s ÞJ~2 ¡21  2 þ 2 c~ SM c~ 1 þ ~ns c~ SM =2 þ ð~n  n0 Þ2 c~ CM

¡22

0 0 J~2 n~  ð~n  n~ s ÞJ~2 þ c~ CM c~ 1 þ ~n2s c~ SM =2 þ ð~n  n0 Þ2 c~ CM

ð31Þ ð32Þ ð33Þ

It follows from eq 29 that ¤cSM ¼ A1 et=tr1 þ A2 et=tr2 and ¤cCM ¼ B1 et=tr1 þ B2 et=tr2 ð34Þ where coefficients A1 , A2 , B1 , and B2 are determined by the initial conditions. Specific slow relaxation times tr1 and tr2 in eqs 12 and 13 can be solved by replacing the parameters ¢ik with ¡ik . The eqs 24­33 generalize the previous theory.1,3,4,8,12,13 In the cases of only spherical or only cylindrical micelles, the results can be obtained by setting cCM ¼ 0 or cSM ¼ 0. This work was supported by the program “Chemistry and Physical Chemistry of Supramolecular Systems and Atomic Clusters” of RAS and the Program of Development of St. Petersburg State University (project No. 0.37.138.2011). Paper based on a presentation made at the International Association of Colloid and Interface Scientists, Conference (IACIS2012), Sendai, Japan, May 13­18, 2012. References 1 E. A. G. Aniansson, S. N. Wall, J. Phys. Chem. 1974, 78, 1024. 2 R. Zana, in Dynamics of Surfactant Self-Assemblies: Micelles, Microemulsions, Vesicles, and Lyotropic Phases in Surfactant Science Series, ed. by R. Zana, Taylor & Francis, Boca Raton, 2005, Vol. 125, Chap. 3, p. 75. 3 A. K. Shchekin, A. P. Grinin, F. M. Kuni, A. I. Rusanov, in Nucleation Theory and Applications, ed. by J. W. P. Schmelzer, Wiley-VCH, Berlin-Weinheim, 2005, pp. 312­374. doi:10.1002/ 3527604790.ch9. 4 F. M. Kuni, A. I. Rusanov, A. K. Shchekin, A. P. Grinin, Russ. J. Phys. Chem. 2005, 79, 833. 5 M. S. Kshevetskiy, A. K. Shchekin, J. Chem. Phys. 2009, 131, 074114. 6 G. Porte, Y. Poggi, J. Appell, G. Maret, J. Phys. Chem. 1984, 88, 5713. 7 S. May, A. Ben-Shaul, J. Phys. Chem. B 2001, 105, 630. 8 F. M. Kuni, A. K. Shchekin, A. I. Rusanov, A. P. Grinin, Langmuir 2006, 22, 1534. 9 M. S. Kshevetskiy, A. K. Shchekin, Colloid J. 2005, 67, 324. 10 A. K. Shchekin, F. M. Kuni, A. P. Grinin, A. I. Rusanov, Russ. J. Phys. Chem. A 2008, 82, 101. 11 A. K. Shchekin, F. M. Kuni, A. P. Grinin, A. I. Rusanov, Colloid J. 2006, 68, 248. 12 M. S. Kshevetskii, A. K. Shchekin, F. M. Kuni, Colloid J. 2008, 70, 455. 13 A. K. Shchekin, F. M. Kuni, K. S. Shakhnov, Colloid J. 2008, 70, 244. 14 A. K. Shchekin, M. S. Kshevetskiy, O. S. Pelevina, Colloid J. 2011, 73, 406. 15 A. I. Rusanov, F. M. Kuni, A. P. Grinin, A. K. Shchekin, Colloid J. 2002, 64, 605. 16 A. P. Grinin, A. I. Rusanov, F. M. Kuni, A. K. Shchekin, Colloid J. 2003, 65, 145. 17 A. I. Rusanov, Micellization in Surfactant Solutions, Harwood Academic Publishers, Reading, MA, 1997.

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