Analytical model of critical buckling transition for smectic liquid crystal based on the viscoelastic scaling of coarse-grained molecular dynamics

The buckling transition of smectic liquid crystals (LCs) is important not only as fundamental physics but also for the rational design of devices to make use of their optical and mechanical properties. However, there exists a huge gap between the specific knowledge and universal analytical formulation. We have conducted coarse-grained molecular dynamics (CGMD) simulations with the force field optimized for the description of buckling phenomena including topological defects to link the molecular nature and continuum formulation. The simulations reveal the viscoelastic characteristics where the critical strain and the compression modulus highly depend on the strain rate as well as the number of layers. Therefore, we formulate the scaling model whose coupling constants depend on both strain rate and domain size. The model reproduces the CGMD results as well as experimental and theoretical values in existing literature. Furthermore, we elucidate from this model that the critical buckling behavior is determined by the competition between the suppression of compression-induced flow and the undulation fluctuation of layers. The framework consisting of the CGMD simulation and the scaling model enables us to estimate the buckling characteristics of smectic LCs reflecting their molecular structures in a wide range from the low-frequency regime that can be verified by experiments to the high-frequency regime beyond the reach of it.

Figure 1

Schematic diagram of smectic $A_d$ cell and its constituent 8CB molecule. A single molecule consists of a cyano-biphenyl group made up of three types of CG particles CN, P3, and C6 and an alkyl group with AL particles. Each layer consists of two sublayers to form the double-layered Sm-$A_d$ phase. Molecules are aligned in the opposite directions in the two sublayers, where the weakly bound AL-AL pair is placed outside the layer and the strongly bound CN-CN pair is placed in the center of the layer. The definition of the $xyz$ coordinate of the system, that of layer angle $\theta$, and that of the chevron tip are also shown.

Figure 2

Basic structural and loading states as a function of strain. (a) $W=lnP\left(\theta \right)$ in the buckling phase. The average layer angle ${\theta }_0$ is defined by the layer angle corresponding to the maximum value of $W$. The interfacial free energy $F_{\mathrm{int}}$ is calculated by $F_{\mathrm{int}}=W_{\max }-W_{\min }$, where $W_{\min }$ and $W_{\max }$ are the minima near $\theta =0^{\circ }$ and maxima near $\theta =\pm {\theta }_0$, respectively. (b) Relation between stress $\sigma$ and strain $s$: the empty circles represent $s\text{−}\sigma$ curve, the empty squares represent $s\text{−}{d}^2\sigma /d{s}^2$, and the solid line represents the linear regression line of $s\text{−}\sigma$ in the range of $s<s_{\mathrm{c}}$. We define the compression modulus $B$ as the slope of the linear regression line, and the critical strain $s_{\mathrm{c}}$ as $s$ at the peak of $s\text{−}{d}^2\sigma /d{s}^2$ nearest to the peak of $s\text{−}\sigma$. The extracted values of $B$ and $s_{\mathrm{c}}$ are shown.

Figure 3

Stress-strain curves of the smectic cells under the conditions of $N=4$, $L=14$, 21, $28\mathrm{nm}$, and $\dot{s}=1.78\times {10}^5$ to $28.5\times {10}^5{\mathrm{s}}^{-1}$, where $N$ indicates the number of layers, $L$ indicates the unit cell size, and $\dot{s}$ indicates the strain rate.

Figure 4

Stress-strain curves of the smectic cells under the conditions of (a) $L=14\mathrm{nm}$, (b) $L=21\mathrm{nm}$, and (c) $L=28\mathrm{nm}$ for $N=4$, 8, 12, and $\dot{s}=7.12\times {10}^5$ to $28.5\times {10}^5{\mathrm{s}}^{-1}$, where $N$ indicates the number of layers, $L$ indicates the unit cell size, and $\dot{s}$ indicates the strain rate.

Figure 5

Distributions of layer angle and the corresponding snapshots of buckling states of the four-layered cell of $L=28\mathrm{nm}$ compressed at lower strain rates of $\dot{s}=1.78\times {10}^5{\mathrm{s}}^{-1}$, when the strains are $s=$ (a) 0.085, (b) 0.17, and (c) 0.255, respectively.

Figure 6

Distributions of layer angle and the corresponding snapshots of buckling states of the four-layered cell of $L=28\mathrm{nm}$ compressed at higher strain rates of $\dot{s}=28.5\times {10}^5{\mathrm{s}}^{-1}$, when the strains are $s=$ (a) 0.085, (b) 0.17, and (c) 0.255, respectively.

Figure 7

Strain dependence of the average layer angle ${\theta }_0$ and interfacial energy $F_{\mathrm{int}}$ of a four-layered cell with $L=28\mathrm{nm}$ under the conditions (a) $\dot{s}=1.78\times {10}^5{\mathrm{s}}^{-1}$, (b) $\dot{s}=28.5\times {10}^5{\mathrm{s}}^{-1}$. The critical strain $s_{\mathrm{c}}$ is added in each figure.

Figure 8

Deformation of (a) the four-layered and (b) eight-layered cells at $s=0.253$ with $L=14\mathrm{nm}$ and $\dot{s}=7.12\times {10}^5{\mathrm{s}}^{-1}$. The locations of the chevron tips on each layer are indicated by empty circles.

Figure 9

Layer angle distributions of (a), (b) first, (c), (d) second, and (e), (f) third layer in the four-layered cell at strains of (a), (c), (e) $s=0.171$ and (b), (d), (f) $s=0.253$, respectively. The cell with $L=14\mathrm{nm}$ is compressed with $\dot{s}=7.12\times {10}^5{\mathrm{s}}^{-1}$. The contour lines of $\theta =0^{\mathrm{o}}$ and $\theta =\pm {15}^{\mathrm{o}}$ are labeled to represent the domain boundaries.

Figure 10

Layer angle distributions of (a), (b) second, (c), (d) fourth, and (e), (f) sixth layer in the eight-layered cell at strains of (a), (c), (e) $s=0.171$ and (b), (d), (f) $s=0.253$, respectively. The cell with $L=14\mathrm{nm}$ is compressed with $\dot{s}=7.12\times {10}^5{\mathrm{s}}^{-1}$. The contour lines of $\theta =0^{\mathrm{o}}$ and $\theta =\pm {15}^{\mathrm{o}}$ are labeled to represent the domain boundaries.

Figure 11

Dependence of (a) the compressive modulus $B_{\mathrm{R}}$ and (b) the critical strain $s_{\mathrm{c}}$ on the strain rate $\dot{s}$. Empty circles indicate the data for the number of layers $N=4$ ($N/N_0=1$, $N_0=4$), empty squares indicate those for $N=8$ ($N/N_0=2$), and empty triangles indicate those for $N=12$ ($N/N_0=3$). Three data sets for $L=14$, 21, and $28\mathrm{nm}$ are used for each condition of $\dot{s}$ and $N$. The regression curves fitted by the scaling functions are also shown [cf. Eq. (27) and Table 2 and Table 3]. The dashed line indicates the regression curve for $N=4$, the dotted line for $N=8$, and the dash-dot line for $N=12$, respectively.

Figure 12

Dependence of (a) the compressive modulus $B_{\mathrm{R}}$ and (b) the critical strain $s_{\mathrm{c}}$ on the number $N$ of layers. Empty circles indicate the data for the strain rate of $\dot{s}=7.12\times {10}^5{\mathrm{s}}^{-1}$ ($\dot{s_0}=1.78\times {10}^5{\mathrm{s}}^{-1}$), and empty squares indicate those for $\dot{s}=28.5\times {10}^5{\mathrm{s}}^{-1}$. Three data sets for $L=14$, 21, and $28\mathrm{nm}$ are used for each condition of $\dot{s}$ and $N$. The regression curves fitted by the scaling functions are also shown [cf. Eq. (27) and Tables 2 and 3]. The dashed line indicates the regression curve for $\dot{s}/\dot{s_0}=4$, and the dash-dot line for $\dot{s}/\dot{s_0}=16$, respectively.

Figure 13

Dependence of (a) the compressive modulus $B_{\mathrm{R}}$ and (b) the critical strain $s_{\mathrm{c}}$ on on the cell size $L$. Empty circles indicate the data for the strain rate of $\dot{s}=1.78\times {10}^5{\mathrm{s}}^{-1}$ ($\dot{s_0}=1.78\times {10}^5{\mathrm{s}}^{-1}$), empty squares indicate those for $\dot{s}=7.12\times {10}^5$, and empty triangles indicate those for $\dot{s}=28.5\times {10}^5{\mathrm{s}}^{-1}$. The data sets for the four-layered cell are used for each condition of $\dot{s}$ and $L$. The regression curves fitted by the scaling functions are also shown [cf. Eq. (27) and Tables 2 and 3]. The dashed line indicates the regression curve for $\dot{s}/\dot{s_0}=1$, the dotted line for $\dot{s}/\dot{s_0}=4$, and the dash-dot line for $\dot{s}/\dot{s_0}=16$, respectively.