Domain spacing scaling

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)... 

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. 

A more comprehensive theory for the thermodynamics of semi crystalline diblocks has been developed using self-consistent mean field theory applied to diblocks with one amorphous block and one crystallizable block [20].The amorphous regions were modelled as flexible chains, and the crystalline regions as folded chains. Both monolayers and bilayers of once-folded chains were considered. Expressions were derived for the thickness of the amorphous and crystalline region and the number of folds. The central result is the domain spacing scaling [Eq. (1)]. 

Douzinas KC, Cohen RE, Halasa AF. Evaluation of domain spacing scaling laws for semicrystalline diblock copolymers. Macromolecules 1991 24 4457-4459. 

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. 

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... 

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).

Finally the total domain spacing was found to scale as... 

Because of the complexity of hydrated PEMs, a full atomistic modeling of proton transport is impractical. The generic problem is a disparity of time and space scales. While elementary molecular dynamics events occur on a femtosecond time scale, the time interval between consecutive transfer events is usually 3 orders of magnitude greater. The smallest pore may be a few tenth of nanometer while the largest may be a few tens of nanometers. The molecular dynamics events that protons transfer between the water filled pores may have a timescale of 100-1000 ns. This combination of time and spatial scales are far out of the domain for AIMD but in the domain of MD and KMC as shown in Fig. 2. Because of this difficulty, in the models the complexity of the systems is restricted. In fact in many models the dynamics of excess protons in liquid water is considered as an approximation for proton conduction in a hydrated Nation membrane. The conformations and energetics of proton dissociation in acid/water clusters were also evaluated as approximations for those in a Nation membrane.16,19 20 22 24 25... 

Pattern and process, cause and effect, are scale dependent. Interaction occurs when both are at the same domain of scale in space and time. 

Self-consistent field theory leads to the prediction that the domain spacing, dy for a lamellar semicrystalline diblock, scales as [20] ... 

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