Form factors for different aggregation models of micelles

Physica B 174 (1991) 192-195 North-Holland

Form factors for different aggregation models of micelles P.S. Goyal a, K. Srinivasa R a o a, B.A. D a s a n n a c h a r y a ~ and V.K. Kelkar b "Solid State Physics Division, Bhabha Atomic Research Centre, Bombay 400 085, India bChemistry Division, Bhabha Atomic Research Centre, Bombay 400 085, India

Spherical micelles in micellar solutions often aggregate to form elongated aggregates on addition of salt. This paper reports calculated small-angle neutron-scattering distributions for dilute CTAB micellar solutions for the case of cylindrical and chain-like aggregates. Several chain geometries have been considered.

I. Introduction Micellar solutions often show an increase in their viscosity on addition of a salt, solution of cetyltrimethylammonium bromide ( C T A B ) and sodium salicylate (NaSal) being an example [1, 2]. It is generally believed that increase in viscosity of micellar solution is connected with the fact that on addition of salt, two or more of C T A B micelles either rearrange themselves to form cylindrical micelles or they join together to form a chain (like a necklace) of spheres [3-5]. The chains (or cylinders) are short and straight at low salt concentrations. However, they grow in length and start folding about themselves as the salt concentration is increased. In general, a chain may fold in many ways; fig. 1. gives some examples. A ferrofluid is an example of another system where one could encounter chains of spherical particles [6]. It is of interest to know if a small-angle neutron-scattering (SANS) experiment can be used to distinguish between a cylinder and a chain; it is also of interest to know if the SANS distribution can provide information about the details of the chain foldings. This paper reports calculated SANS distributions from dilute (0.002 M) C T A B micellar solutions* [7]. C T A B micelles are spherical ( R - 25 A) in pure solutions [5]; they aggregate on addition of salt. The * A part of this work was published in ref. [7].

smaller dimension of the aggregate is decided by the length of C T A ion and continues to be equal to R for several different types of aggregates. In the calculations reported below, we assume that

((3)

Straight Chain (K=12)

tc~ 2 R. K

I, (b)

4

Ring Type c h a i n ( K : 1 2 ) ÷

£',4

~L

i (C) FOLDED CHAIN nx=3

(K =27) 4[,=2Rnx F

Fig. 1. Typical shapes of chains for which SANS distributions have been calculated.

0921-4526/91/$03.50 O 1991- Elsevier Science Publishers B.V. (North-Holland)

P.S. Goyal et al. / Form factors for micellar aggregates

K spherical micelles join to form a cylindrical micelle (length Lcy = 4KR/3 and radius R) or a micellar chain (length Lch = 2KR) in presence of NaSal. For the case of a chain, the calculations have been done for (a) straight chain, (b) circular chain, (c) chain having several U-shaped folds and (d) a Gaussian (Debye) chain [8].

193

that constitutes the chain and K, the number of beads in the chain, it can be shown that form factor Pch(Q) of the chain is given by [12]

1 Pch(Q) = ~ Psp(Q) K-1

× K+ Z

r E

sin(Qdij)] j.

(4)

i=1 j = i + l

2. Theory Small-angle neutron-scattering (SANS) experiment measures the coherent differential scattering cross-section dZ/dO as a function of the scattering vector Q. The scattering vector Q is the difference vector between the wavevectors of the incident and scattered neutrons and has a magnitude Q = 4~r sin( ½~b)/A where A is the neutron wavelength and ~b is the scattering angle. In case of dilute micellar solutions, where inter-micelle interference effects are negligible, it can be shown that [9, 10] dX

dO ( o ) = Kn(pc - pD)2vzp(Q) '

Here dij is the distance between the ith and the jth bead. For a given geometry of the chain, the above distances can be easily obtained from the coordinates of the centers of the spheres. In case of a Debye chain, it can be shown that [8]

1

Pc(O)=~Psp(e)

[

I+(K-1) ×--7(x-l+e

-x

,

(5)

X

where x = (QRg) 2 and Rg = lch/V~ is the radius of gyration of the chain.

(1)

3. Results and discussions where n is the number of micelles per unit volume of the pure CTAB solution and v is the volume of the spherical micelle. Pc and PD are the scattering length densities of the micelle material CTAB and the solvent D20. P(Q), which is the square of the form factor of the aggregate, is decided by the shape and size of the aggregate. P(Q) for a spherical micelle of radius R is given by [9] [ 3(sin(QR) - QR cos(QR)) ]z Psp(Q) = / (-Q~3-~ ,

(2)

and that for a cylinder of length tcy and radius R is given by [11] ~r/2

P~y(Q) =

f 0

×

• 2

1

4sin (~QLcyCOSO) z 2 Q L ~y c o s 2 / 3

4J~(QR sin/3) Q2REsin2/3 sin/3 d/3 ,

(3)

where Jl is the Bessel function of first order. If Psp(Q) is the form factor of the spherical bead

SANS distributions have been calculated for 0.002 M C T A B / D 2 0 solutions assuming that K CTAB micelles join to form the chain or the cylinder. Calculations have been done for several values of K both for the cylinder and the chain. In view of a limited Q range (0.001-0.2A -l) available in SANS experiments, aggregates with K > 15 were not considered; very small aggregates (K < 3) were also not considered as they are not of much interest. Figure 2 shows the calculated distributions for K = 15 corresponding to cylindrical micelles, straight micellar chains, circular chains and Debye chains. It is seen that there are significant differences in the calculated cross-sections for a cylinder and a chain. The distributions from different geometries of the chain are similar at large Q, but they show differences in the low-Q ( < 0 . 0 4 ) region. It was seen that the distributions for the folded chains (fig. lc) always matched with the distribution for the straight chain at large Q > 0.07 A - l, irrespective of the values of n x and ny; the distributions are sensitive to nx and ny at low Q, but the

P.S. Goyal et al. / Form factors for micellar aggregates

194

3.0 ~

CTAB(0.002 M)

2.5 ill ii/i II ~! !I ~ 2.0 li I

0,0 0.0

curves are mostly between the corresponding curves for a straight chain and a cylinder. Coming to distinguishing between a cylinder and a straight chain, we note that the crosssections for the cylinder are larger than those for the chain over the complete Q region. Figure 2 shows the results for K = 15; similar differences have been seen for other values of K. It is thus possible to distinguish between the two shapes if the same number of micelles join to form a cylinder or a straight chain. It is not clear at this stage if, for a given K, the SANS distribution from a cylinder can match with the distribution from a chain for a different K. T o answer this question, we refer to fig. 3 where SANS distributions for chains and cylinders for several K values are shown. We note that both for cylinders and chains, the crosssections are independent of K for Q > 0.05 ~-1. This is clearly seen in the inset in fig. 3; the upper boundaries of the shadowed regions correspond to K = ~ and the lower boundaries correspond to K = 3. Thus we find that irrespective of the value of K, d , ~ / d O for a cylinder is larger than that for a chain in the large-Q region. This fact can be fruitfully exploited for distinguishing between a cylinder and a straight chain even if they are of different lengths. In suitable cases it

CYLINDER STRAIGHTCHAIN RINGTYPECHAIN --x-- DEBYE CHAIN K=15

i

i

i

i

0.0:~

0.04

0.06

0.08

O-I

Q(A ) Fig. 2. Calculated SANS distributions for 0.002 M C T A B D20-salt solutions corresponding to different aggregation models of micelles. To maintain the clarity of the figure, the results for folded (fig. 1(c)) chains are not shown in this figure (see text for discussion).

Z+[

I

I

I

_I

!

}

CTAB (O.OO2M) L K =12

--

\~

CyLinder

--choio

-~,..~

I 0.0

~

7

K;~6 \

6

\

I

I

1 I I I 0.02 0,04 o-1 WAVE VECTOR TRANSFER Q ( A )

I

I

I 0.06

I

I

~

I

008

Fig. 3. Calculated SANS distributions for 0.002 M CTAB-D20-sait solutions for different sizes (K) of the aggregates. The solid lines are for cylinders and dashed for chains. Large-Q data are shown in the inset. The lower boundaries of the shadowed region in inset correspond to K = 3 and the upper boundaries correspond to K = oo.

P.S. Goyal et al. / Form factors for miceUar aggregates

should be possible to obtain their lengths also as it is seen that even in the low-Q ( < 0 . 0 3 A -1) region, none of the dashed lines in fig. 3 coincides with any of the solid lines. The above discussion is strictly true only for monodisperse system. We had confined ourselves till now to straight chains. The fact that large-Q cross-sections are independent of the shape of the chain (fig. 2) and the fact that large-Q cross-sections can be used to distinguish between a straight chain and a cylinder suggest that a SANS experiment can be used to distinguish between a chain and a cylinder, irrespective of the shape of the chain. However, it may not be always possible to obtain the detailed shape of the chain if the value of K is not known. The present calculations show that different chain geometries can give similar distributions in the Q range of SANS experiments for different values of K. The above discussion is for dilute solutions; at high concentrations, one will have to take account of the inter-micelle interference effects.

195

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