Copolymer volume

Fig. 3.20 Fraction, , of copolymer chains aggregated in micelles as a function of the overall copolymer volume fraction, (j>. for different values of the incompatibility parameter yN (a - 5, N — 200) (Leibler etal. 1983).The number of copolymer chains in a micelle, p. and the volume fraction of copolymer monomers in the micelle corona, y, depend weakly on . Full line, y/V = 20, for

Fig. 6.1 (a) Electron micrograph showing micellar aggregates in a blend of a PS-PMMA diblock (A/w = 175kgmol1, /ps0.53) in a blend with PS homopolynier (A/w = 95 kg mol 1) with a copolymer volume fraction c = 0.3, at room temperature (Lowenhaupt and Hellmann 1991). (b) Enlargement of a micellar aggregate. R denotes the solvent evaporation rate. 

Fig. 6.23 Logarithmic plots of the correlation length ( ) and zero-angle scattering intensity (/(0)) as a function of temperature reduced with respect to the Lilshitz temperature Tlp) for a blend of PE and PEP homopolymers with a PE-PEP diblock (details as Fig. 6.22) at a copolymer volume fraction

Fig. 6.36 Phase diagram calculated using SCFT for a blend of a symmetric diblock with a homopolymer with fl = 1 (see Fig. 6.32 for a blend with a diblock with / = 0.45) as a function of the copolymer volume fraction

We first consider the special case of a copolymer melt, for which the overall density (i.e., copolymer volume fraction) is constrained to take everywhere its maximum value p0 — 1. The free energy (87) then simplifies to... 

We now consider the random copolymer model in the presence of solvent— that is, for a copolymer volume fraction p0 < 1. We are not aware of previous work on this model in the literature, but will briefly discuss below the link to models of homopolymer/copolymer mixtures [57]. The excess free energy (86) then depends on two moment densities, rather than just one as in all previous examples. For simplicity, we restrict ourselves to the case of a neutral solvent that does not in itself induce phase separation this corresponds to X = 0, making the excess free energy... 

Figure 13. Coexistence curve for cooolymer with solvent the parent has a uniform density distribution p (o) = const. = /2 with overall density (i.e., copolymer volume...

As an example, consider a uniform parent distribution p(°) (a) = const, which can be written as p (o) = p /2, using the fact that pj, = do p (a). Then pf 1 — pff / 3 and p =p /5 and so a tricritical point occurs if the overall density (i.e., the copolymer volume fraction) is pf = 3/(2r + 3). Figure 13 shows the coexistence curve calculated for this parent (with r = 1), which clearly shows the tricritical point at the predicted value X1 = l/(2/4° ) = r + 3/2 = 2.5. Our numerical implementation manages to locate the tricritical point and follow the three coexisting phases without problems we take that as a signature of its robustness [58]. Note that the tricritical point that we found is closely analogous to that studied by Leibler [57] for a symmetric blend of two homopolymers and a symmetric random copolymer that is, nonetheless, chemically monodisperse (in the sense that o = 0 for all copolymers present). In fact, in our notation, the scenario of Ref. 57 simply corresponds to a parent density of the form p (o) S(o — 1) + S(o +1), with the copolymer (o = 0) now playing the role of the neutral solvent. 

Figure 6.19. Interfacial tension between polystyrene (relative molecular mass 4000) and poly(dimethyl siloxane) (relative molecular mass 4500) as a fimction of the copolymer volume fraction in the PDMS phase. The relative molecular mass of the styrene block of the copolymer is 5500 and that of the dimethyl siloxane block is 7500. The interfacial tension in the absence of block copolymer was 4.85 0.05 mJm. After Hu et al. (1995).

The effect of the micelles on the interfacial tension is to limit the possible reduction. This is shown in figure 6.18, in which the interfacial tension has fallen to less than one third of its initial value by the time the copolymer volume fraction has reached 0.05. But the adsorption isotherm, figure 6.17, reveals that this is the critical micelle composition and above this composition the chemical potential becomes a very weak function of the copolymer volume fraction and very little further reduction in interfacial tension is to be expected. 

FIGURE 19.7. Interfacial activity in a Type I blend of an A-B diblock copolymer added to a blend of A and B homopolymers [A = SPB(89) and B = SPB(63)]. A/a = 4,230 and A/b = 3,600 for the homopolymers, while A/Ab = 790 and Nsb = 730 for the block copolymer. Symbols show experimental measurements using secondary-ion mass spectrometry (SIMS), and curves show SCFT predictions using x and / values from Tables 19.1 and 19.2. (a) Volume fraction profile in loglinear format of the diblock copolymer for a sample with 0.07 volume% block copolymer with an A/B interface at z = 190 nm. (b) Volume fraction profile in linear-linear format of the diblock copolymer for a sample with 0.07 volume% block copolymer with an A/B interface at z = 190 nm. The cross-hatched area represents the adsorbed amount, F. (c) Adsorption isotherm the dependence of the adsorbed amount, r, on the copolymer volume fraction in the A-rich phase ab/a- (d) The thickness of the adsorbed layer (standard deviation of the volume fraction profile near the peak), a, plotted versus the amount adsorbed, F. 

However, the conditions under which micelles form are rather different from the conditions under which copolymers microphase separate, for example, Eq. (1). These conditions are described by the cmc, which can be defined as the minimum copolymer volume fraction needed to form micelles. To a first approximation, this model predicts... 

EVA copolymers are the most important on a volume basis. The total LDPE copolymer market for Europe is estimated at 720 kt/jT. The EVA copolymers volume is 655 kt/yr of which 450 kilotoimes are above 10 wt-% VA). 

The porous copolymer volume often is higher than the liquid phase volume (consisting of monomers and inert medium) because the network monomers adsorption on the swollen network determines the network s volume expansion. 

Mayaduime RTA, Rizzardo E. Mechanistic and practical aspects of RAFT polymerization. In Jagur-Grodzinski J, editor. Living and Controlled Polymerization Synthesis, Characterization and Properties of the Respective Polymers and Copolymers, Volume 65. New York Nova Science Publishers 2005. 

The important features of the approximate relationships (96) and (98) were verified by the exact numerical calculations. An exponential dependence of the interfacial tension reduction on the block copolymer molecular weight as well as on the total homopolymer volume fraction was predicted that can explain the remarkable effectiveness of using large molecular weight diblocks as surfactants for concentrated mixmres of immiscible homopolymers. For small N, a linear dependence of Ay on N (98) was also predicted by the exact numerical calculations. Moreover, a linear dependence of Ay on the block copolymer volume fraction was predicted by the exact numerical solution, as shown by (96) and (98). 

The interfacial tension increment. Ay = y yo, was linear with the copolymer volume fraction, calculated for low concentration of the copolymer additive as suggested by theory for concentrations below the CMC. The slope of the fitted line was —37.0, and thus d was estimated to be 38 nm, or 63.5b when the geometric mean of the Kuhn statistical segment lengths of the two segments was used as 0.6 nm. This value of d ( 63.5 monomer units) was about 24% of the contour length of the copolymer chains and, thus, indicated an extended configuration of the copolymer chains. 

Fig. 29 Interfacial tension increment (Ay) versus copolymer volume fraction c) for the PS/PS-h-PVE/PVE system at 145°C. Solid line is the linear fit of the data for concentrations below the CMC, according to the theory of Noolandi and Hong. From [45]...

Here y N is the degree of incompatibility of the species, k-g, is the Boltzmann constant, T is the absolute temperature, 4> is the average copolymer volume fraction, and b is the Kuhn statistical segment length. Formally, the same expression for yo would be obtained if there were no copolymer chains in the system. 

Figure 4.1 Variation of the number of transition via chain inser-tion/expulsion (a) micelle merger/sphtting (b) and micelle spanning (c) with the copolymer volume fraction (top = Nb = 10 e = 0.45) and the pairwise interaction parameter e bottom-, Na = Nb = 10 < ) = 0.05). Number of Monte Carlo steps 35,482. Reproduced from Reference 22 with permission of the American Chemical Society.

AB) star copolymers (volume fraction of B block/ = 0.2) have a conformational structure similar to that of star polymers. 

---