(χN)OD increases with decreasing film thickness when the film thickness is less than 6L0 due to the substrate compatibility effect of the neutral underlayer to BCP films. The phase behavior of high molecular weight BCPs converges with the results of mean field theories.
Experimental Results
In this way, each polymer is represented by N + 1 segment positions, but practically we can consider that the number of segments is N; ie 21.29] It is common to choose δ(r) =δ(r) in standard field-based simulations because the range of segment interaction is negligible for high molecular weight polymers.
Single Chain in External Fields
The average segment concentration of the K-type block in the αth chain is an ensemble average of the instantaneous segment density ˆφα,K, and this can be obtained by a functional derivative of the partition function of the single chain (2.17) with respect to WK with using the above relationship. For example, the entire equations can be reproduced by using the bond distribution function of the freely connected chain model.
Many-Chain Polymer System
The Dirac delta functional can be written as a product of many one-dimensional delta functions when the lattice number M goes to infinity. Since the inverse pair potential u−1(r−r0) is isotropic, we can combine the first and the second term in equation (2.40) and express it as a functional of. which does not contain W±, the effective Hamiltonian can be written as βH[W−, W+] = ρ0. 2.43).
Saddle Points of Hamiltonian
Considering equations (2.37) and (2.44), the saddle point of the pressure field always lies on the imaginary axis, and the path with constant phase lies parallel to the real axis. Thus, by moving the contour path to pass through the saddle point while remaining parallel to the real axis, the value of the Hamiltonian can be obtained more easily because the integral is dominant near the saddle point. 61] In Chapters 3 and 4, we present field-based simulations for computing the canonical partition function in the vicinity of saddle points.
As mentioned earlier, our goal now is to perform an integral in the vicinity of the saddle point. One method that is widely used is to apply the saddle point approximation to both W− and W+ fields. 3.2) By using the saddle point approximation, the integral can be approximated analytically, without ignoring fluctuation of fields.
Discrete Chain Self-consistent Field Theory
In the standard SCFT, a segment at position only interacts with other segments that share the exact same position. The DCSCFT can be formulated in the same way, but there is an alternative choice, suggested by Matsen, who introduced non-bonded interactions of finite range. 51] In the current study, we use the Gaussian finite range function u(r), which represents the interaction strength between two segments at distance r.
While the change of the two fields, wAext(r)−wBext(r) does not change, the summation of the fields wextA (r) +wBext(r) only affects the free energy reference points. To calculate the surface interaction, we choose the following form of wBext(r). Note that the externally imposed wall interactions located at the two interfaces propagate through the convolution integral, which is consistent with the idea of finite-range interaction between segments; that is
Methods
- Pseudo-spectral Method
- Iteration Scheme
- Neutral Boundary Condition
- Hard-wall Boundary
- DCSCFT of BCP with mat
- Identifying the ODT location
One way to obtain the distribution function is to adopt direct sampling of the analytic function in Fourier space. Instead, we choose to sample the distribution function in real space so that it has a finite cutoff length. For BCPs in the thin film geometry, we impose hard walls on the top and bottom surfaces.
In the actual implementation of this method, we use the discrete cosine transform, taking into account the discreteness of the computational space and the symmetry of the binding function. To identify the ODT location, we analyze the normalized density variation intensity in Fourier space, and it is calculated from the 2D discrete Fourier transform of the segment concentration contrast. The analysis in Fourier space provides the structural information of the system and allows us to identify the transitions compared to experimental ODTs.
Results and Discussion
- Cylinder-forming BCPs Confined within Preferential Surfaces
- Lamella-forming BCPs Confined within Preferential Surfaces
- Lamella-forming BCPs Confined between Preferential Surface and Neu-
- Lamella-forming BCPs Confined within Neutral Surfaces
We investigate the ODT behavior of the BCP films confined between a weakly preferential surface and a neutral base mat (denoted as single-neutral) and between two neutral surfaces (denoted as double-neutral). Note that even when χN is very low, a symmetry of the system in the z direction is already broken by the preferred top surface. In order to isolate the effect of the neutral bottom substrate, it is desirable to neutralize the top surface by also setting ηtop = 0.
As the film thickness increases to 30L0, the (χN)ODT of the films confined in the doubly neutral state decreases significantly until the thickness reaches 8L0, but thereafter it gently decreases to the bulk value at ∼22L0 or thicker. Similarly, the (χN)ODT of the films confined in the en-neutral state rapidly decreases to the bulk value at 5L0 and reaches a minimum at ∼10L0. The dashed line shows (χN)ODT. b) Plot of (χN)ODT for double- and single-neutral films with BCP thickness from 2L0 to 30L0.
Conclusion
Introduction
One of the main challenges in CL-FTS is the instability of the simulation, which limits the size of the time step ∆τ and allows only a very small time step. 47] A disadvantage of this smearing interaction is the loss of universality of the standard Gaussian chain, which means that renormalization and calibration methods based on the standard Gaussian chain model cannot be used. Alternatively, the instability problem due to the complex nature of the effective Hamiltonian can be eliminated by performing a partial saddle point approximation on pressure fields.
In this partial saddle point approximation method, functional integral over pressure field W+ is evaluated using the saddle point approximation, while only the exchange field W− fluctuates. It appears that this partial saddle point approximation method does not suffer from severe instability that is problematic for the CL-FTS. Recently, it has been demonstrated that L-FTS is much better than MC-FTS in computational speed when running simulations with a relatively large simulation box and small ¯N, since the amplitude of the MC motions in the MC-FTS is limited in this regime.
Methods
Langevin Field-Theoretic Simulation
In the first few iteration steps, the pressure field is updated using a simple Euler relaxation method. After a few iteration steps, when it is within the AM threshold, we switch to Anderson's mixing method to update the pressure field. In this Anderson mixing method, we calculate the aforementioned inner product (Eq. (3.31)) using only the pressure fields.
If the incompressibility error becomes smaller than the tolerancetor = 10−4, we abort the iteration for the pressure field. The difference is that now only the pressure field is updated in this loop, while both W± are updated in the SCFT. When we find the saddle point of the pressure field, we escape from the nested loop.
Pseudo-spectral Method
Ranjan, Qin and Morse proposed a 4th-order pseudo-spectral method that applies the Richardson extrapolation to the 2nd-order pseudo-spectral method. 77,84] As long as we assume the Gaussian distribution as the bond distribution g(r) and choose Dirac delta function as pair interaction potential au(r), we can use the 4th-order pseudo-spectral method proposed for the continuous Gaussian. chain. The numerical calculation of equation (4.17) is essentially the same as the pseudo-spectral method case described in subsection 3.3.1.
The errors are determined by subtracting the free energy value from the exact value obtained from the SCFT calculation with N = 1000 and 4th-order accuracy pseudo-spectral method. The blue and red lines are the errors of 2nd- and 4th-order pseudospectral methods, respectively. To satisfy the target segment density near the walls, the pressure field in those regions grows rapidly, and many saddle point iterations are required to satisfy the incompressibility condition and achieve the stability of simulation.
Ultraviolet Divergence and Renormalization of Flory-Huggins Parameters 50
To exploit this renormalization method, we use a cubic grid with the same grid spacing regardless of the size of the simulation box. The dashed line is the prediction of the RPA theory and the solid line is the result of the ROL theory. All these plots obtained from L-FTS with different χN parameters in the disordered phase.
The structure function obtained without PT shows severe metastability, while (right) the peak value of the structure function with PT varies monotonically with respect to χbN. The peak value of the PT-free structure function does not vary monotonically with respect to the χbN parameter, which means that severe metastability is present in the data. By performing L-FTS with PT, the peak value of the structure function varies monotonically with respect to the χbN parameter, indicating that the PT method greatly promotes the formation of a stable well-ordered morphology.
Results
Cylinder-forming BCPs Confined within Neutral Surfaces
59] For the location of ODT points, the standard method is to calculate the order parameter according to equation (4.26). The story is almost the same for the double film, except that (χN)ODTis 13.8, and the transition to the cylindrical phase occurs at χN = 14.5. The χ/χb ratio is calculated to be 0.924 using RPA renormalization, which means that the effective χN value is from 16.7 to 17.6. e) Scheme of order parameters as functions of χN.
This increase is more pronounced for the monolayer film, and thus the monolayer enters the ordered phase at a relatively higher (χN) value. Figure 4-10 shows that the uncertainty for (χN)ODT due to the hysteresis effect is within 0.1 for the cylinder-forming polymers with ¯N = 8×103. (χN)ODT increases as ¯N decreases, and the increase is more pronounced for the monolayer case compared to the bilayer case at the experimentally relevant ¯N value. e) Plot of order parameters as functions of χN.
Discussion and Conclusion
Helfand, Theory of inhomogene polymers: fundamentals of the gaussian random-walk model.The Journal of Chemical Physics. Mahmoudi, Segregation of chain ends to the surface of a polymer melt.The European Physical Journal E. Matsen, Calibration of the Flory-Huggins interaction parameter in field-theoretic simulations.The Journal of Chemical Physics.
Matsen, Efficiency of pseudo-spectral algorithms with Anderson mixing for SCFT of periodic block copolymer phases. The European Physical Journal E. Fredrickson, A multispecies exchange model for fully fluctuating polymer field theory simulations. Journal of Chemical Physics. Dorfman, Open source code for self-consistent field theory calculations of block polymer phase behavior on graphics processing units. European Physical Journal E.
