Using the delta correction to the standard free energy \cite{bc} in the elastic setting with a quadratic foundation term and some parameters, we introduce a one dimension only model for strong segregation in diblock copolymers, whose sharp interface periodic microstructure is consistent with experiment in low temperatures. The Green's function pattern forming nonlocality is the same as in the Ohta-Kawasaki model. Thus we complete the statement in [31,p.349]: ``The detailed analysis of this model will be given elsewhere. Our preliminary results indicate that the new model exhibits periodic minimizers with sharp interfaces.'' We stress that the result is unexpected, as the functional is not well posed, moreover the instabilities in $L^2$ typically occur only along continuous nondifferentiable ``hairs''. We also improve the derivation done by van der Waals and use it and the above to show the existence of a phase transition with Maxwell's equal area rule. However, this model does not predict the universal critical surface tension exponent, conjectured to be 11/9. Actually, the range $(1.2,1.36)$ has been reported in experiments [21,p. 360]. By simply taking a constant kernel, this exponent is 2. This is the experimentally ($ \pm 0.1$) verified tricritical exponent, found e.g., at the consolute $0.9$ K point in mixtures of ${}^3$He and ${}^4$He. Thus there is a third unseen phase at the phase transition point.
Deep Dive into Asymptotically periodic L^2 minimizers in strongly segregating diblock copolymers.
Using the delta correction to the standard free energy \cite{bc} in the elastic setting with a quadratic foundation term and some parameters, we introduce a one dimension only model for strong segregation in diblock copolymers, whose sharp interface periodic microstructure is consistent with experiment in low temperatures. The Green’s function pattern forming nonlocality is the same as in the Ohta-Kawasaki model. Thus we complete the statement in [31,p.349]: The detailed analysis of this model will be given elsewhere. Our preliminary results indicate that the new model exhibits periodic minimizers with sharp interfaces.'' We stress that the result is unexpected, as the functional is not well posed, moreover the instabilities in $L^2$ typically occur only along continuous nondifferentiable hairs’’. We also improve the derivation done by van der Waals and use it and the above to show the existence of a phase transition with Maxwell’s equal area rule. However, this model does not pr
Key to some recent developments in modern microelectronics has been the ability to create high-quality atomically abrupt interfaces between different semiconductors, which can produce new quantum states and uncover unexpected phenomena (see http://en.wikipedia.org/wiki/Nanotechnology). Several experiments contradicting common knowledge about bulk materials having only diffuse interfaces have been reported, see the References, especially [7], where a new experimental technique is explained.
It is remarkable that chemically diverse diblock copolymers and microemulsions admit very similar characteristic structures, e.g., the bicontinuous ordered double diamond [27]. A theory formally predicting the phase transition between weak and strong segregation was derived in [4]. Here we show how to use elasticity and choose parameters in the Vaserstein pseudoassociation potential to get local L 2 minimizers resembling the square wave [38]. As a by-product we get a rigorous proof of the gas to liquid phase transition.
To make this note self-consistent, we review the history of this modeling. The ideal gas law (0.1)
where P is the internal pressure, V the volume per mole, T the absolute temperature, R the gas constant, does not predict a phase transition between gas and liquid, defined according to experiment as two densities coexisting at the same pressure (see e.g., the isotherms for carbon dioxide in [36]).
Van der Waals formulated the modified equation of state as (0.2)
, where a is the attraction parameter which arises from polarization of molecules into dipoles and b is the volume enclosed within a particle (the repulsion parameter). Below the critical temperature isotherms for (0.2) have a wiggle [36], which is unphysical. Maxwell’s construction in which the fluid is taken around a reversible cycle of states from a logical point of view is worthless [36], as states on the wiggle have no meaning. However, such soft reasoning can give other interesting results [22,36].
For uniform systems, the Helmholtz free energy is defined as Ψ = U -T S, where U is the internal energy and S the entropy. The first law of thermodynamics states dU = δQ -P dV , where δQ is heat change, the second law for reversible systems states δQ = T dS. Thus dΨ = dU -T dS -SdT = δQ -P dV -T dS -SdT , so P = -∂Ψ ∂V and Ψ = -P dV . Since the isothermal compressibility
and the convex envelope of (0.3) gives Maxwell’s equal area rule for the isotherms of (0.2). However, this and related statistical mechanics arguments lack a proof of phase transition as defined above. To be more precise, suppose that Maxwell’s envelope is on the line segment [V 1 , V 2 ]. Why does the system take on only V 1 and V 2 , but not other values inside the segment? In other words, the implicit meaning of arguing this way is that assuming there is a phase transition, it is determined by the equal area rule.
As is often the case in mathematics, we can get a proof of phase transition with Maxwell’s equal area rule by studying a related larger structure. This was done in [20, p. 40]. In a related work, the author also obtained 4/3 as the critical exponent [21, p. 360]. Other methods have reported 3/2 (van der Waals), 1.38 [28] and (1.21, 1.32) [42].
We now improve the derivation of van der Waals. Let ρ = 1 V denote the density. The free energy of a nonuniform state ρ(x) of a nonreversible process is a functional of the Helmholtz form I(ρ) = U (ρ) -T S(ρ). The cumulative free energy
U (ρ) cannot be only
Cρ can be neglected due to the mass constraint, one can conclude that (0.5)
Note that -aρ 2 -RT ρ(ln(ρ -1 -b)) is concave (double-well) for ρ and T satisfying RT ≥ (<) 2aρ(1-ρb) 2 . However, note that the nonlocal and the -1 0 aρ 2 terms appeared quite separately, therefore it may not be entirely convincing that one is an extension of the other. Also, if -1 0 aρ 2 is replaced by -1 0 1 0 J(x -y)ρ(x)ρ(y)dxdy, the solutions of the EL equation are always continuous, so we have no phase transition as defined above.
In his Nobel lecture, van der Waals expressed his absolute conviction that molecules associate in complexes not of chemical origin. He called them pseudoassociations. Thus the derivation can be improved by adding the nonlocal term to (0.4):
The mass constraint makes possible an addition of a linear term, which we choose so that -RT ρ ln(ρ -1 -b) -aρ 2 has equal depth wells. Also, on a bounded interval boundary effects may come into play, so J is not necessarily translationally invariant: J = J(x, y). Now (0.6) is qualitatively the same as derived in [4] (0.7)
where W is the double-well function
with j(x) = 1 0 J(x, y)dy and k the Boltzmann’s constant. For some works on this and related models see the References, especially the collective diffusion kinetics of the phase transition in polymer gels, what the authors in [24] consider one of the most exciting problems in current condensed matter physics, and a nonlocal in time evolution discussed in [37].
Let
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