Solvents athermal

The simultaneous agreement of exponents in Eqs. (12) and (14) characterizes the crossover condition. Then it is derived that the validity of Eq. (14) corresponds to N(j>>f 3 . This means that for an athermal solvent, where 3=1, the intermediate region governed by Eq. (12) disappears, while for a theta solvent Eq. (14) is not applicable. 

Generation 6 Internal poor solvent Terminal athermal solvent normal micelle ... 

Generation 6 Internal athermal solvent Terminal poor solvent loopy micelle ... 

However, the surface coverage is the same for both copolymers when weakly adsorbed to the surface. Surface density profiles were also compared. Finally, scaling relationships for triblock copolymer adsorption under weak adsorption conditions were derived (Haliloglu et al. 1997). In a related paper (Nguyen-Misra et al. 1996), adsorption and bridging of triblock copolymers in an athermal solvent and confined between two parallel flat surfaces were studied, and the dynamic response of the system to sinusoidal and step shear was examined. 

A more detailed explanation of Eqs 3 and 4 shall be given in section 2. For our present purposes, the main conclusion is that Jc is not dependent on X for linear chains. But, without any calculation, we can expect that a branched object will enter much more painfully in a narrow tube. Thus permeation studies may provide an interesting assessment of branching. In section 3, we discuss briefly the case of star polymers [11] which is relatively easy to visualise. Then in section 4, we study the morerealistic problem of branched objects [12]. All our discussion is very crude restricted to polymers in very good (athermal) solvents and restricted to the level of scaling laws. 

The first term is an elastic energy, for a molecule of extension L along the tube, with an unperturbed radius R0 given by Eq.(2). The second term describes monomer/monomer repulsions (in an athermal solvent). Optimising Eq.(14), we arrive at... 

It is instructive to consider some limiting cases of [3.4.56]. For monomers (N = 1) in an athermal solvent [x =0), only the logarithmic term remains the sum reduces to the term for z = 1 (trains). For dilute solutions (0 1), we thus obtain =fcrin(l-0j), which is identical to the Langmuir expression [3.4.34] with 0 = 6(1) = 0. For monomers in a poorer solvent (where multilayers may form due to attraction between the monomers) the layers z > 1 also give a contribution, according to the last term on the r.h.s. of [3.4.56[. 

In a typical case of the Mayer /-function with an attractive well, repul-sion dominates at higher temperatures and attraction dominates at lower temperatures. In athermal solvents with no attractive well there is no temperature dependence of the excluded volume. It is possible to have monomer-solvent attraction stronger than the monomer monomer attraction. In this case, there is a soft barrier in addition to the hard-core repulsion and the excluded volume v>b d decreases to the athermal value v = b d at high temperatures. 

In order to emphasize the difference between ideal and real chains, we compare their behaviour under tension. Consider a polymer containing A monomers of size b, under tension in two different solvents a (9-solvent with nearly ideal chain statistics and an athermal solvent with excluded volume An ideal chain under tension was alreacly discussed in... 

The thennal blob size is the length scale at which excluded volume becomes important. For v the thermal blob is the size of a monomer Ht b) and the chain is fully swollen in an athermal solvent [Eq. (3.12)]. For V —b, the thermal blob is again the size of a monomer ( 7- Ri b) and... 

End-to-end distance of dilute polymers in various types of solvents, sketched on logarithmic scales. In a 0-solvent the thermal blob size is infinite. For athermal solvent and non-solvent the thermal blob is the size of a single monomer. Good and poor solvents have intermediate thermal blob size (shown here for the specific example of equivalent thermal blobs in good and poor solvent. 

The two functions are compared in Fig. 3.18. Note the dramatic difference between them. Real chains in an athermal solvent rarely have ends in close proximity. The probability to find chain ends within relative distance dx of X is 47tx P(x) dx. The coefficients of the distributions of end-to-end dis-... 

In an athermal solvent, the monomer-solvent energetic interaction is identical to the monomer-monomer interaction. This makes the net interaction between monomers zero, leaving only the hard core repulsion between monomers. The excluded volume is independent of temperature (v Ks b ), and the chain is a self-avoiding walk of monomers ... 

Why is there no temperature dependence of the excluded volume in an athermal solvent ... 

The crossover volume fraction is of order unity (f) 1) in an athermal solvent, meaning that the chains are partially swollen at all concentrations. The excluded volume in an athermal solvent is fully screened only in the melt state (. rs b 7- at 0 = 1). 

Figure 5.9 compares osmotic pressure in a 0-sol vent (filled circles) with osmotic pressure in the same solvent 25 K above 0 (filled squares) and in an athermal solvent (open symbols). The 0-solvent data clearly exhibit the slopes of 1 and 3 expected from Eq. (5.56). At 0 + 25 K, the solvent is a... 

This equation holds for theta, good, and athermal solvents. Hence, osmotic pressure or osmotic compressibility measurements provide a con-venient means ot measuring the correlation length in semidilute solutions. 

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