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Effect of micellar flexibility on the isotropic-nematic phase transition in solutions of linear aggregates
T. Odijk
To cite this version:
T. Odijk. Effect of micellar flexibility on the isotropic-nematic phase transition in solutions of linear aggregates. Journal de Physique, 1987, 48 (1), pp.125-129. �10.1051/jphys:01987004801012500�. �jpa- 00210413�
Effect of micellar flexibility on the isotropic-nematic phase transition
in solutions of linear aggregates
T. Odijk
Department of Physical and Macromolecular Chemistry, Gorlaeus Laboratories, University of Leiden, 2300 RA Leiden, The Netherlands
(Reçu le 23 juin 1986, accepté le 25 septembre 1986)
Résumé. 2014 On considère le couplage entre la croissance et l’alignement de micelles linéaires. Si les micelles sont
rigides, ce couplage aboutit à une croissance presque illimitée des micelles dans la phase nématique. Nous montrons
que la prise en compte d’une flexibilité réaliste des micelles réduit considérablement ce couplage. Nous dérivons des
expressions analytiques des variables de coexistence pour toute la gamme de flexibilités. Une augmentation du paramètre de croissance des micelles modifie leurs propriétés intrinsèques et abaisse le paramètre d’ordre de la phase nématique à la transition.
Abstract. 2014 If linear micelles are considered to be rigid, their growth couples so strongly to alignment in the nematic
phase that they grow almost without bound but we show that this coupling is greatly diminished when a realistic
flexibility is taken into account. Analytical expressions for the coexistence variables are derived for the complete
range of semiflexibility. Enhancement of growth by modifying the intrinsic properties of the micelles leads to a
decrease in the order parameter of the nematic phase at the transition.
Classification
Physics Abstracts
36.20 - 61.30 - 82.70
1. Introduction.
In developing the statistical mechanics of micellar
aggregates we have no other option but to focus on the
chemical potential > of the amphiphiles themselves.
Hence, end effects which generally play only a minor
role in the physics of unalterable particles, now domi-
nate the expressions that result on minimizing > with
respect to s, the number of amphiphiles within a micelle [1]. In particular, there exists an intriguing coupling
between the growth and the alignment of micelles in the nematic phase because, in that case, u contains a term stemming from the orientational entropy of align-
ment which depends on both s and the variational parameter a expressing the sharpness of the orienta- tional distribution function. Gelbart, McMullen and Ben-Shaul [2-4] have developed such a theory for
rodlike and disklike aggregates that are perfectly rigid.
The object of this paper is to investigate the effect of flexibility on their calculations for unidimensional (i.e.
rodlike) micelles.
The first systematic theory of the isotropic-nematic
transition for semiflexible or wormlike polymers was
formulated only recently by Khokhlov and Semenov [5, 6]. Their treatment can also be understood in terms of a
scaling analysis [7] (for an exhaustive review on the
ubiquitous influence of flexibility on a variety of liquid- crystalline systems, see Ref. [8]). For highly confined
wormlike polymers the deflection length [7-9]
A = P/a is the important scale rather than the persist-
ence length P (a = 0 ( 10 ) ; A P). When they are longer than about A, semiflexible chains cannot be
approximated by rods. They must then be viewed as
curves deflected towards the director about once every A. Thus, the semiflexibility effect shows up for very short chains. Its incorporation in detailed calculations leads to an almost quantitative explanation of exper- imental results on very stiff biopolymers, results that cannot be rationalized by using rigid rod models [8].
It stands to reason that micellar flexibility must also
be important in the description of micellar nematics.
Here, we outline a theory within the second virial
approximation but we must bear in mind several severe
restrictions when trying to relate theoretical conclusions to experiments. Higher virial coefficients can be disre-
garded only when unidimensional micelles are
anisometric enough. However, their degree of elonga-
tion is not known precisely [10-14]. Furthermore, the
micelles are considered to be wormlike cylinders in- teracting via hard core repulsions. This is surely an oversimplification of these complex systems. In ad- dition, the persistence length is poorly specified exper-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004801012500
126
imentally because interacting micellar solutions are
semidilute. Estimates of P range from 20 to 104 nm [15- 21]. The interpretation of the viscosity measurements has been criticized [22] but the (approximate) applica- bility of a fully rigid rod model does not rule out a
persistence length of the same order as the contour length. Moreover, figure 5 of reference [21] confirms
the semiflexibility of one micelle. Anyhow, an a priori evaluation of the micellar persistence length on the
basis of elasticity arguments gives P =103 nm [28].
Most micelles are already flexible according to our
criterion (contour length L=0(A); A = P / a =
10-100 nm as will be seen below).
2. Coexistence equations.
We write the chemical potential A (in units kg 7J of an amphiphile in solution as follows, assuming the only
allowable aggregates are monodisperse and wormlike
This is identical with the form proposed by Gelbart
et al. [2] (see their Eq. (7)) except for the present
incorporation of the influence of flexibility in the
orientational entropy term 03C3. Equation (1) splits up
naturally into five terms : 1) a constant uo ; 2) an
excluded-volume effect proportional to the volume fraction cp and an orientational factor p, and inversely proportional to the minimum aggregation number m (the constant w of order unity will be specified later on) ; 3) end effects promoting growth and inversely proportional to the aggregation number s ; R = K + ln cp + ke cp with K an intramicellar growth
constant left unspecified in this work, ke cp arising from
end effects in the excluded volume interaction and In cp signifying a mixing entropy term ; 4) terms pro-
portional to In s arising from mixing (-1), decrease in rotational degrees of freedom - 3 2 and decrease
in translational ones -11 - - 6 ; (- 2 5) ) the orien-
tational entropy of confinement proportional to o-.
Let us postulate two possible phases, one isotropic (denoted by index i), the other nematic (index n), so
one can further specify the form of equation (1).
I) For the isotropic solution we have simply
II) For the nematic solution we have
The dimensionless excluded volume p, proportional to
the average of the sine of the angle y between two
infinitesimal segments, is calculated with the help of the
orientational distribution function f ( D ) depending
on the solid angle 12 = (8, lp) defined with respect to the director. We will allow f to be a function also of one
parameter a which is to be determined variationally. In
R we have set k., i == k,,,,, = 1 which proves to be an
excellent approximation (compare with Ref. [8]).
Finally, for semiflexible chains, even the formal ex- pression [6] for a. is unwieldy ; but for the purpose of this paper we show that the orientational entropy separates to a good approximation into intensive and extensive parts.
Once we have specified the free energy, the isotropic
to nematic transition is analysed by using the Onsager recipe [23]. The parameters s and a are found by minimizing A with respect to s and IL n with respect to s
and a
The prime superscript denotes differentiation with respect to a.
The solution si, s,, --+ oo feasible only if cr, 1= 0 will
be discussed later on. Equality of the chemical poten- tials in the respective phases together with equations (1, 4, 5) lead to
We also require the osmotic pressure to be the same in both phases. We calculate the osmotic pressure from
the Helmholtz free energy AF which is obtained from
N
equation (1) : OF =
Jo
N uskB T dN + Z ( T, s ) whereZ does not depend on V because
and N is the number of micelles.
There are five coexistence equations (4-8) determin- ing five variables : ’Pi’ cpn, Si’ sn and a. We can discover
a way of solving them by considering firstly two limiting
cases : micelles that are completely rigid and rodlike,
and those that are very long and semiflexible.
3. Rigid rodlike micelles.
In equations (3, 6 and 7) we set a, = 0. It is possible to disentangle equations (4-8) by introducing the new
variables : length ratio q = sn/ Si’ concentration ratio A = ’Pn/ ’Pi’ width Ag = cpn - cpi, 8 hg,, - qgi and
the slowly varying combination B = - p n a’ p’. The
set (4-8) now boils down to having to solve only three equations
The presence of Ocp and 5 appears to be a nuisance but
Ao 1- h-1 and 5 h - qh-1 so they are perturba-
tive and can be deleted on a first iteration.
We have tried solving equations (9-11) with 5 and Ao = 0, using the trial functions proposed by Onsager [23]
It turns out that the left hand side of equation (9) is
smaller than the right hand side unless either q = 1,
h = 1, and a = 0 (i.e. the isotropic state) or q, h and a
are exceedingly large. Numerical work shows that the 5 and Ao perturbations do not change this conclusion.
But it is not definitive. The 6 term arises from the interaction between micellar caps and bodies so that its numerical value depends on the micellar shape. Furth-
ermore, higher order virial coefficients also contribute and are of the same order of magnitude (when
~p,, is no longer much smaller than unity). Of course
these deliberations have no bearing on the fact that the
coexistence quantities of Gelbart et al. [2], based on setting cp = cp n, change dramatically when, instead, equation (8) is taken into account (see Eq. (18) ; note
that forcing ’Pi equal to cp n in this work yields coexist-
ence quantities differing from those of Ref. [2] because
the respective numerical coefficients in A differ slight- ly).
The very highly ordered state can be assessed analyti- cally by using a Gaussian trial function
We have [22]
i. e . B = 2, so that equations (9-11) with 5 and Ao = 0
reduce to
Accordingly, we obtain the only feasible nematic solution described by
JOURNAL DE PHYSIQUE. - T. 48, N- 1, JANVIER 1987
(This solution is more stable than the previous sn’ si -+ oo solution by 4 kB T per micelle).
The solution (Eq. (18)) is evidently meaningless.
From equation (18) one can gather that the persistence length must be some 1012 times longer than the
diameter if the rigid rod model is to be valid, an untenable premise. Furthermore, the virial expansion
breaks down for very high a (to be specific when
a 1/2 is larger than the inverse axial ratio [8]). Let us
then turn to the opposite limit.
4. Very long semiflexible micelles.
Because we expect high a values, it is plausible to use
the Gaussian approximation again (Eq. (14)). Actually,
the calculation of the orientational entropy of a con- fined semiflexible chain is a very hard one and few
128
results are known [5-9]. Fortunately, the Gaussian approximation is tractable for all contour lengths and
its explicit dependence on a is very accurately given by [8]
where M = LlP is the number of persistence lengths in
a micelle (L = contour length, P = persistence length).
When we set M equal to zero, we regain the rigid rod
limit (Eq. (15)) exactly if the implicit expression for o-is
used (see Ref. [8]) and almost exactly for o- given by equation (19). The high M limit yields
The reader will recognize the extensive term from references [5-7]. A detailed discussion of equations (19, 20) can be found in reference [8].
It is easy to see that we can let the contour length Ln become as long as we want, simply by chosing a large enough growth parameter K. Thus, the case Ln > P is well defined and we denote it by the index 0.
It is convenient to introduce the micellar diameter D which lets us define the constant w, viz,
= 20132013 =0(1) Ds (1) and the small parameterp
11 = D/P. Taking the formal limit L or s -+ oo in
equations (6, 7, 8) fand using equation (20) we are left
with a very simple set of expressions
Using equations (4, 5 and 16) we find the solution valid for Ao = 0
a 0 = 81 / iT = 25.8 (order parameter So == 0.88)
This solution is stable (note that there is no sj, Sn --+ 00
solution). It will be very useful in devising approxima-
tions as in the next section. Equations (22) are easily
amended when Ocp is not completely negligible since it
is only a perturbation.
5. Intermediately long micelles.
When the length of the micelles is in between the rigid
rod and the fully semiflexible limits, equations (4-8) are conveniently rewritten as a function of the number of deflection lengths M À = aM
(Again, dcp = 0 and 6 = 0).
For the experimentally interesting case Ma ? 5, equation (24) is well-approximated by
where we have used the Gaussian trial function again.
This approximation yields equation (22) exactly and
does not stray too far from equations (10) and (11).
Hence equation (23) gives
Moreover, equations (24) and (25) immediately yield
Noting that equations (5) and (6) provide another
relation between Ma and a
we can draw the following conclusions : at the transition
a = a (,q’e K ) and similarly for h, cpn etc. ; the
nematic order decreases when growth is promoted by increasing K ; enhancement of the persistence length
leads to an expected increase in a ; an important parameter is ( K + 7 In q ) ; various micellar systems should exhibit similarity in phase behaviour if they can
be dealt with by the present model. The approximation
Ao = 0 in equation (9) and the replacement of equation (19) by equation (20) have little or no effect
on these conclusions. We also note that equations (23- 25) describe a stable nematic state. A final remark
concerns the behaviour of the nematic order and micellar length beyond the volume function cp n: these variables simply increase monotonically.
6. Discussion.
When micelles are modelled as slender, rigid, impenetr-
able cylinders and constrained to be monodisperse in
the way originally proposed by Gelbart et al. [2-4], the
micelles grow almost catastrophically as the nematic
phase is reached. Gelbart et al. did not reach this conclusion because they approximated the concentra-
tion ratio h by unity, in this way deleting the equation expressing the constancy of the osmotic pressure. At first sight, this approximation looks sound enough but equation (9) shows that h is very strongly coupled to the length ratio q. A virtually runaway state ensues. In
addition, it is straightforward to prove that slightly
nematic states do not exist and we have also found no
evidence for the feasibility of nematic states with
a = 0 ( 10 ) . Contrastingly, Gelbart et al. [3] do find a
solution for weakly nematic solutions provided the
micelles are allowed to be polydisperse. Further investi-
gation into the influence of polydispersity is obviously
warranted. It is stressed that for highly ordered polydis-
perse rods, re-entrant behaviour must be reckoned with
as shown numerically [24] and analytically [25].
These considerations about the stabilization of mono-
disperse rodlike nematic micelles become somewhat academic once the effect of semiflexibility is incorpo-
rated into the theory. Here, we have proved that explosive growth is prohibited because of the elemen- tary fact that persistence lengths are never kilometers
long. The physical reason for the strong inhibition of
growth is that the free energy of a microsystem like a
micelle must ultimately scale as its size when it is large enough. The isotropic-nematic transition involves inten- sive quantities so these must level off at some stage.
One bothersome feature of previous models and also this one is that the micelles are predicted to keep on lengthening when the concentration increases. There is indirect evidence that micelles may diminish in size in semidilute solutions [26]. Moreover, the powerful tech- nique of magnetic birefringence reveals that when
growth does occur, it is not quantitatively understood [27]. Hence, equation (1) will surely be modified in the future. Unfortunately, there are many interactions that
are viable candidates since end effects dominate in
equation (1). These difficulties aside, the simple calcu-
lations outlined here may prove useful in interpreting
the behaviour of those micellar systems that can be shown to exhibit unequivocal growth of the aggregates.
Acknowledgments.
The author thanks Henk Lekkerkerker for several discussions.
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[28] The wormlike chain model is equivalent to an elastic
rod in a heat bath ; the persistence length P is proportional to the bending force constant 03B5 of the rod. Since 03B5 is in turn proportional to the
farth power of the diameter and linear micelles
are about twice as thick as double-stranded DNA, we arrive at P for micelles ~103 nm
because P (DNA) ~50nm. This implicitly as-
sumes the elastic properties of organic matter do
not vary by more than an order of magnitude.
