Extracci´ on del contorno de larva

Spherical micelles appear in solution when the polymer concentration exceeds the so- called critical micelle concentration (CMC). The CMC is specified by the following conditions: (i) the chemical potential µ0 of a free copolymer (unimer) in solution coincides with the chemical potential µ of copolymer molecule in the equilibrium micelle; (ii) when the translation entropy of micelles is neglected, the chemical potential µ is equal to the free energy per molecule, µ=F3(Q) where F3(Q) is the free energy per molecule in a spherical

micelle with aggregation number Q. Assuming that a unimer constitutes a spherical globule of collapsed block B and swollen block A, we find

) 1 ( ) ln( ₃ 0 =kT c +F Q= µ (54)

where ln(c) is the translational entropy of unimer in solution with concentration c. Then the CMC for spherical micelles is determined as

[

]

c_CMC) ( ) ( 1) /

Unlike in micelles where the insoluble block is stretched, in unimer the collapsed block is compressed with respect to its Gaussian size. The corresponding deformation free energy of the insoluble block in the core of a unimer can be written as

3 2 3 1 2 0 2 0/ B B B / B B B ϕ B kT a N p R p N F ≅ ≅ (56)

The free energy of the swollen soluble block is composed of only a single blob and as such . Therefore the leading contribution to the free energy of a unimer in solution, F

1 /

0 kT ≅

F_A

3(Q=1), is due to the surface free energy Fs0 as long as F_B0 << F_s0. This then means that the CMC can be written as

[

]

c_CMC) ( ) _s /

ln( ≈ ₃ − ₀ (57) Using equation 10 for Fs0

(

)

As seen from equation 58, both polymer and SMWS micelles exhibit a CMC. However, since NB for polymer micelles is much larger, the CMC for polymer micelles is going to be much lower than for SMWS micelles. In addition, unlike small molecular weight micelles which go from micelles to unimers over a narrow concentration range, polymer micelles often have a broad concentration range over which they go from micelles to unimers caused by the polydispersity of NB.74 The longer the insoluble block the larger the drive for the diblock to form micelles, and thus the lower the CMC.49,50 Meaning that in the CMC region diblocks with longer insoluble blocks have started to form micelles while as those with shorter insoluble blocks still exist as unimers in solution. As seen in equations 57 and 58, the CMC is directly related to temperature. Further more, as will be discussed in more detail in section 8.1 changes in temperature can also affect γ. This means that as the temperature

changes, the CMC also changes. The temperature when micelles form (at a constant concentration) is called CMT.

Since the aggregation number changes with concentration as diblocks begin to form micelles, it is impossible to extrapolate the scattering intensity to zero concentration. Thus, only an apparent aggregation number can be measured in the CMC region.73 The

variation in the apparent aggregation number with concentration for two different polymer samples is shown in Figure 20. To determine the CMC from these plots, the classical definition of the CMC: the concentration at which micelles are first detected, i.e. when the aggregation number is higher than 1, is used

to determine our CMC values. Since the intensity of light scattered is proportional to the concentration, very low CMCs are difficult to measure. For samples 39-15 and 39-26 it was only possible to measure the initial decrease in the aggregation number. This made it necessary to extrapolate the change in aggregation number with concentration in order to obtain the CMC. Since molecules with smaller soluble blocks

Figure 20. The change in the aggregation

number with concentration for PS-b-PI 39- 94 (black squares) and 19-99 (grey circles).

Figure 21. Plot in the variation of the CMC

with the molecular weight of the PI blocks. Black and grey symbol’s are for series 2 (40kDa PS) and series 3 (20kDa PS) respectively. Solid lines are results of the analytical solution (eq 58).

have a larger gain in free energy when they form micelles, the CMC should decrease as the PI block is reduced. Figure 21 and Table 3 show the variation in the CMC with the molecular weight of the PI block. In addition Figure 21 shows the theoretical predictions on how the CMC (eq 58) changes with the molecular weight of the PI block. As seen in the figure, the theoretical and experimental points for series 3 (20kDa PS) are in close agreement, while theory predicts CMC’s are about three orders of magnitude lower than those experimentally measured for series 2 (40kDa PS). We believe that this discrepancy is partially due to the 40kD series being out of equilibrium at 25°C as was discussed in section 6.1. Since the CMC is sensitive to temperature, being kinetically trapped at a higher temperature would elevate the CMC helpin between the theoretical and experimental data.

Table 3: Experimentally Measured CMC Values Series 1 sample CMC dCMC 40-12 3.07E-08 1.74E-08 40-13 2.95E-08 1.45E-08 40-17 1.26E-07 6.78E-08 Series 2 sample CMC dCMC 39-15 1.5E-08 6.2E-09 39-25.6 2.2E-07 1.5E-08 39-52 3.2E-07 2.2E-07 39-94 7.2E-07 1.3E-07 Series 3 sample CMC dCMC 20-13 1.75E-06 1.26E-07 20-14 3.27E-06 3.22E-07 20-19 1.89E-06 2.67E-07 20-26 3.08E-06 3.79E-07 20-59 2.20E-05 1.00E-06 20-99 5.07E-05 4.74E-06

Experimentally measured critical micelles concentrations (in g/ml) for micelles formed from series 1, 2, and 3 diblocks. The furthest right column (dCMC) gives the error range for each measured value.

g to explain some of the discrepancy