Domain spacing

The comparison of a linear PS-fr-PI-fr-PS triblock with its linear analogue was performed by Takano et al. [97]. As in the diblock case the cyclic copolymer exhibits a smaller domain spacing however, the effect is not so pronounced (Table 3). This finding may be explained by the tendency of the ABA triblock to exhibit a higher curvature compared to an AB diblock, which in consequence reduces the differences between a cyclic diblock and its linear ABA counterpart. 

The results for different domain spacings are summarized in Table 3. It is obvious that cyclic block copolymers exhibit smaller domain spacings compared to their linear analogues, which is due to an entropically less favourable situation and an increased interfacial curvature. 

Fig.49 Composition distribution dependence of reduced domain spacing, D/D0, of PS- -P2VP with single microphase-separated structure. Do = 60.8 nm is domain spacing of parent copolymer with Mn = 125 kg/mol. Hatched region macrophase separation. From [160], Copyright 2003 American Chemical Society...

Initially Klepeis et al. allowed the dihedral angles to vary over the entire [—7T, 7r] domain. It was found, however, that the problem required intensive computational effort (Androulakis et al., 1997). A reduction of the domain space was therefore proposed by setting limits based on the actual distributions of the dihedral angles. Obviously, for the algorithm to be successful, these reductions could not exclude the region of the global minimum conformation. 

Fig. 2.5 Schematic showing the variation of inverse scattering intensity and domain spacing (as determined from SAXS or SANS) across the order-disorder transition of a block copolymer melt. The mean field transition temperature has been identified operationally as the point where, on heating, the inverse intensity crosses over to a linear dependence on T (after Sakamoto and Hashimoto 1995).

Minimization of the total free energy (sum of eqns 2.2 and 2.3) with respect to d leads to a predicted domain spacing scaling as (Semenov 1985)... 

The expressions 2.7-2.12 which define the Leibler structure factor have been widely used to interpret scattering data from block copolymers (Bates and Fredrickson 1990 Mori et al. 1996 Rosedale et al. 1995 Schwahn et al. 1996 Stiihn et al. 1992 Wolff et al. 1993). The structure factor calculated for a diblock with / = 0.25 is shown in Fig. 2.39 for different degrees of segregation JV. Due to the Gaussian conformation assumed for the chains (Leibler 1980), the domain spacing in the weak segregation limit is expected to scale as d Nm. 

Fig. 2.56 Beating of Kiessig fringes observed using X-ray reflectivity from a PVP PS-PVP tri block copolymer film (fes = 0.48, Mv = 120 kg mol ) with two discrete thicknesses of 1935 and 2229 A (de Jeu et al. 1993). The difference in height results from island and hole formation at the free surface, and is equal to the bulk domain spacing.

For a selective solvent, a scaling relation for the domain spacing in the ordered lamellar phase d (l/T)1 3 was obtained from SAXS experiments on a PS-PB... 

A systematic study of the domain spacing scaling in two nearly symmetric PS-PI diblocks in neutral solvents was also undertaken by Hashimoto et al. (1983b). Results of S AXS experiments on these polymers dissolved in toluene and dioctyl phthalate (DOP) were summarized in a scaling relationship for the domain spacing in the ordered phase in the semidilute and concentrated regimes (0.15 < 4> < 0.6)... 

Crystallization in poly(ethy ene)-poly(ethylethylene) (PE-PEE) semicrystalline diblock copolymers has been investigated using SAXS and WAXS on oriented specimens. Microphase separation was found to precede crystallization for all samples, with 37-90wt% PE (Douzinas and Cohen 1992). The scaling of the lamellar domain spacing in the crystalline phase for the same samples was determined from measurements of the principal SAXS peak position (Douzinas et al. 1991). It was found that the domain spacing scales in agreement with the predictions of the theory of Whitmore and Noolandi (1988) (Section 5.3.5), i.e. 

The domain spacing obtained by Hamley et al. (1997a) and Ryan et al. (1995) increased discontinuously upon crystallization, as indicated by the shift of the principal peak position, q, to lower q, as apparent in Fig. 5.3. Here q = AnsinOIX where 20 is the scattering angle and X is the X-ray wavelength. The SAXS profiles from the crystallized diblocks were shown to correspond to the sum of scattering from block copolymer lamellae, with up to four orders of reflection, plus a broad... 

Fig. 5.6 Small-angle scattering invariant (line), domain spacing (d, open circles) and PE lamellar thickness (/, filled circles) versus time during a series of quenches for a PE-PEE diblock (M = 20kgmol, /pE = 0.55) (Hamley etal. 1997a). The sample was successively quenched from 140°C to 104 C, 100°C, 98°C, 106°C. (Tm (PE) = 108°C for this sample.)...

Crystallization in asymmetric diblocks with compositions = 0.35 and 0.46 was also investigated by Hamley et al. (19966). It was found that a lamellar structure melted epitaxially (i.e. the domain spacing and orientation were maintained across the transition) to a hexagonal-packed cylinder structure in the /PE = 0.35 sample. This is illustrated in Fig. 5.15, which shows SAXS patterns in the solid and melt states, with a schematic of the epitaxial melting process (Hamley et al. 1996a.b). The same epitaxial transition has been observed for a polyethylene oxide)-poly(buty)ene oxide) diblock (Ryan et at. 1997) vide infra). 

Fig. 5.19 Domain spacing (top line) obtained from SAXS for copolymer PEO75PBO54 as a function of time during the melting and crystallization programme indicated by the lower line (Mai el al. 1997). The melt has a gyroid structure.

Fig. 5.30 Scaling of lamellar domain spacing (reduced by N) with the degree of polymerization of the amorphous block, Nt, for various semicrystalline diblocks (Nojima et al. 1995). ( ) Data for PE-PEP diblocks from Rangarajan et al. (1993) (o) data for a PCI. PDMS PCL triblock from Lovinger et al. (1993) ( ) data for PCL-PB diblocks from Nojima et al. (1995).

Measurements of the lamellar domain spacing by a number of groups (Douzinas et al. 1991 Nojima et al. 1995 Rangarajan et al. 1993 Unger et al. 1991) as a function of Nc yielded exponents consistent with eqn 5.24. However, this comparison assumes that d = 4 which is not usually even a good approximation, thus the agreement seems fortuitous. 

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