Ultraviolet Divergence and Renormalization of Flory-Huggins Parameters 50

4.2 Methods

4.2.3 Ultraviolet Divergence and Renormalization of Flory-Huggins Parameters 50

4.2 Methods

4.2.3 Ultraviolet Divergence and Renormalization of Flory-Huggins Param-

4.2 Methods

one-loop (ROL) theory, [57] where the S(k) is the structure function defined as [58,59]

S(k) ρ₀N = np

V² D

δφˆA(k)δφˆA(−k)E

= np

(V χ_b)² hW₋(k)W₋(−k)i − 1 2χ_bN

= N¯^1/2V a³N^3/2

1

(V χ_b)² hW₋(k)W₋(−k)i − 1

2χ_bN, (4.22)

where δφˆ_A(r)≡φˆ_A(r)−f and the Fourier transform is defined as ˜g(k) ≡R

drg(r) exp(ik·r).

The nonlinear fitting relation is suggested as follows χ= z_∞χ_b+c₁χ²_b

1 +c2χb

, (4.23)

where c₁ and c₂ are fitting parameters. It is believed that this Morse calibration is the state- of-the-art renormalization technique known today and it works even for homopolymer mixture unlike the other known renormalization techniques. [90] However, the attemp to apply the ROL theory to asymmetric diblock copolymers (i.e.f 6= 1/2) has not been successful [57,91,92], and thus it is not applicable to the cylinder-forming BCPs.

For the cylinder-forming BCPs, we instead use the renormalization method proposed by the Vorselaars and Matsen based on random phase approximation (RPA) theory. Their renormal- ization equation applicable for asymmetric diblock copolymers is [59,93]

χ=

1−la²NR

S_{RP A,0}(k)dk (2π)³ρ0N f(1−f)

χ_b, (4.24)

whereSRP A,0 is the structure function of RPA evaluated at χN = 0, and its integral is limited within the cutoff frequency. [23] This expression reduces to equation (4.19) at large ¯N. [59] In the mean field limit ¯N → ∞, the bare Flory-Huggins parameter reduces to the effective Flory- Huggins parameter, thus unlike FTS, the SCFT results become accurate as the grid spacing converges to 0. In order to utilize this renormalization method, we use a cubic grid with the same grid spacing regardless of the size of the simulation box. This also makes it somewhat meaningful to compare χ_b directly even without using renormalizations.

For the purpose of comparing renormalization methods based on the RPA and Morse cali- bration, we perform L-FTS of Gaussian chain model for symmetric (f = 0.5) diblock copolymers with ¯N = 1024. The Langevin step ∆τ is set to 0.01 and the number of simulation grid is set to 32 in each direction with periodic boundary conditions. Simulation data are collected at every 10∆τ after the system reaches an equilibrium. In order to get accurate results at lowχN values, the predictor-corrector algorithm [54, 94] is utilized, and the continuous version of equations (2.21) and (2.22) are evaluated using the Simpson quadrature. Because the simulation is per-

4.2 Methods

i,j and kare integers.

Figure 4-2shows the structure functions obtained from L-FTS at variousχ_bN values corre- sponding to the disordered phase. TheS(k) is the average value of structure function atk=|k|. AsχbN increases, the peak value of the structure function increases. For each curve, S(k) ap- proaches to 0 as k goes to 0, which indicates that the L-FTS is producing accurate structure functions. Figure4-3 showsρ0N/S(k^∗) as functions of χN after applying various renormaliza- tion methods. The peak values are estimated using the polynomial regression. The dashed line is the prediction of the RPA theory and the solid line is the result of ROL theory. The blue dots denote ρ₀N/S(k^∗) as a function of χ_bN. The red and green dots denote the results of Morse calibration and RPA renormalization, respectively. Whichever method we choose, the renormal- ized value is much closer to the theoretical predictions, and the Morse calibration exhibits a slightly better match. The simulation result using partial saddle point approximation method must reduce to that of RPA theory at smallχN, and this is confirmed by the observation that the red dots lie on the dashed line when χbN ≤7.0. [59,54]

0 2 4 6 8 10

0 0.1 0.2 0.3 0.4

kaN ^1/2

S ( k ) /ρ 0 N

χ

_b

N = 4.0 χ

_b

N = 7.0 χ

_b

N = 10.0 χ

_b

N = 13.0

Figure 4-2: Structure function of symmetric diblock copolymer with ¯N = 1024. All of the these plots obtained from L-FTS with variousχN parameters in the disordered phase. AsχbN parameter increases, the peak values of the structure function increases.

4.2.4 Parallel Tempering

In the field-based simulation, the final results depend strongly on the initial fields, and there is a hysteresis that prevents the swift change of densities even under the presence of large fluctuations. It is also observable that highly metastable defects often disrupt the formation

4.2 Methods

0 5 10 15

0 5 10 15 20

χN ρ

0

N /S ( k

∗

)

ROL RPA χbN

RPA Renormalization Morse Calibration

Figure 4-3: Comparison of Morse Calibration and RPA Renormalization. The dashed line is the prediction of the RPA theory and the solid line is the result of ROL theory. The blue dots denote ρ0N/S(k^∗) as a function ofχbN. The red dots denotes the results of Morse calibration, while green dots denote the results of renormalization method based on RPA theory.

of a well-ordered morphology. Such phenomena can disturb one to find the accurate ODT point from the simulation. To overcome the hysteresis and reduce meta-stable states, a parallel tempering (PT) method is adopted in this research. [58,95]

To find an ODT point, we run multiple simulations in parallel with uniformly increasingχ_b parameters separated by interval ∆χb. Each simulation has a process number from 1 toSP T in ascending order of χb, that is{χ⁽¹⁾_b , χ⁽²⁾_b , . . . , χ^(S_b ^{P T)}}. For every interval of 10³∆τ, we attempt to swap the field configurations of two adjacent χb values,W_±⁽ⁱ⁾ and W_±⁽ⁱ⁺¹⁾, with probability

pswap = min (

1,exp

"

ρ₀

χ⁽ⁱ⁾_b − ρ₀ χ⁽ⁱ⁺¹⁾_b

! Z dr

W₋⁽ⁱ⁾(r)2

−

W₋⁽ⁱ⁺¹⁾(r)2#)

(4.25)

= min (

1,exp

"

a³ρ0

N¯^1/2 a³N^3/2

N

χ⁽ⁱ⁾_b − N χ⁽ⁱ⁺¹⁾_b

! Z dr

W₋⁽ⁱ⁾(r)2

−

W₋⁽ⁱ⁺¹⁾(r)2#) ,

wherea³ρ₀is the relative segment density. Since pressure fields are chosen to satisfy zero average, R drW₊⁽ⁱ⁾(r) = 0, the pressure fields do not appear in this equation. When swapping the two simulations, we swap bothW₊⁽ⁱ⁾ andW₋⁽ⁱ⁾at the same time. The swapping is tried in a way that the odd and even indicesiare chosen alternatively in each PT attempt.

4.2 Methods

2.5 3 3.5 4

10⁰ 10¹

kaN

^1/2

S ( k ) /ρ

0

N

No parallel tempering

2.5 3 3.5 4

10⁰ 10¹

kaN

^1/2

χbN = 28.2 χ_bN = 28.3 χbN = 28.4 χ_bN = 28.5 Parallel tempering

Figure 4-4: Structure functions obtained from L-FTS withf = 0.3, ¯N = 1536 and variousχbN values. (left) The structure function obtained without PT shows severe metastability, whereas (right) the peak value of the structure function with PT changes monotonically with respect toχbN.

To check the efficiency of PT, we run L-FTS with f = 0.3 and large fluctuation ¯N = 1536 for 4×10⁵∆τ, where ∆τ is chosen to 0.2. Ensemble average of structure function from 10⁵∆τ to 4×10⁵∆τ with interval 1000∆τ is shown in figure4-4. The peak value of the structure function without PT does not change monotonically with respect to the χ_bN parameter, which means that severe metastability is present in the data. Performing L-FTS with PT, the peak value of the structure function changes monotonically with respect to theχ_bN parameter, showing that the PT method greatly promotes the formation of a well-ordered stable morphology.

21 21.5 22 22.5

0 10 20

χN

Ψ

Initial fields : Disordered phase Initial fields : Lamellar phase

Figure 4-5: The order parameters of symmetric diblock copolymers with ¯N = 1024 obtained by L-FTS. For both curves, simulations are conducted with PT method and the initial configurations are set to (blue) the disorder phase and (red) the lamellar phase.

4.2 Methods

However, even with the adopt of PT, the hysteresis is not completely removed when ¯N is small and the fluctuation is strong. [54, 59] Figure 4-5 shows order parameters of symmetric diblock copolymers with ¯N = 1024 obtained from L-FTS utilizing the PT method. The order parameter is defined as suggested by Beardsly et al. [54]

Ψ≡ N

V 2

maxk kW₋(k)²k. (4.26)

The red and blue lines are simulation results whose configurations are initialized with lamellar and disordered phases, respectively. Even though the PT method is utilized, (χN)_ODTestimated by the order parameter turns out to depend strongly on the initial configuration. At these parameter values, the uncertainty of (χN)ODT turns out to be approximately 1. In order to find the precise (χN)_ODTvalue at small ¯N, we need to employ a more sophisticated free energy calculation method such as the thermodynamics integration method (TIM). [47, 96, 97] The TIM is computationally expensive, and thus it is not easily applicable to the relatively large simulation box we adopt in this research. Fortunately, the hysteresis is negligible in our BCP film systems, so uncertainty of (χN)_ODT is lower than its dependence on the layer thickness.