Computational Analysis to Optimize the Performance of Thin Film Liquid Crystal Biosensors

Abstract

A nonlinear unsteady-state mathematical model employing torque balance and Frank free energy according to the Leslie-Ericksen continuum theory is developed and implemented to simulate the performance of nematic liquid crystal biosensor films with aqueous interfaces. A transient liquid crystal-aqueous interface realignment is modeled using the Euler–Lagrange equation by changing the easy axis when the surfactant molecules at the interface are introduced. In our study, we evaluated the dynamics between bulk and interface by controlling surface properties of the interface, such as homeotropic anchoring energy and surface viscosity. In addition, transient optical interference and response time have been examined in this study. Our parametric study results indicated that both homeotropic anchoring energy and surface viscosity at the interface contribute to bulk reorientation. Furthermore, the obtained numerical results indicate that as homeotropic anchoring strength increases, the effective birefringence decreases more gradual due to the increasing surfactant concentration at the aqueous interface, consistent with available experimental observations. Our results have been validated and compared to experimental results from thin-film liquid crystal biosensors in this study.

1. Introduction

Biosensors consist of a combination of receptors and transducers that detect biological or chemical analytes. As the receptor interacts with the target analyte, the transducer converts the recognition process into a quantitative signal. Hence, biosensors can provide signals related to a molecule’s concentration, enabling them to identify biological or chemical agents in the environment [1,2]. The application of liquid crystals in biosensors is relatively new in biotechnology [3,4,5,6]. Previous studies demonstrated that liquid crystal sensors respond to different physical stimuli such as temperature, electric and magnetic fields [7].

The studies in the last few decades have shown that liquid crystals can be used as the basis for new types of biosensors to detect a wide range of chemical and biological targets rapidly, making them useful for portable analytical devices. As a result, several liquid crystal biosensor platforms have been developed and studied experimentally [1,8,9,10]. For example, the use of surfactant-decorated liquid crystal surfaces to detect biomolecule binding [11].

In the absence of external fields such as electric or magnetic fields, when liquid crystal comes into contact with other isotropic phases, reorientation of the director through the bulk can be explained by molecular interactions at the interface as a result of surface and physicochemical properties. Moreover, the reorientation of the bulk due to interface alignment depends on elastic constants, surface anchoring, and bulk and surface viscosities which can lead to the equilibrium state by attaining the balance between bulk and surface [12,13].

In order to study the response of liquid crystal to chemical analytes, the approach consists of evaluating changes in the liquid crystal bulk orientation and response due to the surface realignment that has been investigated experimentally [14]. Furthermore, several studies show that the concentration of chemical or biological targets can control the anchoring of liquid crystals on the aqueous phase [13,15,16].

Hence, adsorbate-driven transitions at the interface can be the basis for liquid crystal biosensors. The thermotropic liquid crystal is in contact with an aqueous phase that is immiscible with water, forming a stable liquid crystal-aqueous interface. The adsorption of chemical analytes by liquid crystals at the interface causes realignment of the liquid crystal at the surface. In the presence of the sufficient amphiphilic concentration, it results in a transition from the initial planar alignment to the homeotropic which causes the bulk to be reorientated [1,11,17]. Second, the detection mechanism of liquid crystals biosensors is mainly based on changes in optical properties of liquid crystals, such as birefringence. This process is illustrated in Figure 1. Therefore, the optical signal is used to observe the changes in the liquid crystal bulk orientation associated with the transition from planar to homeotropic alignment at the surface by examining the created texture and optical pattern when light is transmitted through the deformed system [2,8,18].

Researchers have conducted several experiments [1,8] to optimize liquid crystal biosensors with an aqueous interface. For example, recent studies investigated the effects of liquid crystal elastic constants and surface anchoring energy on the sensitivity of biosensors. According to these studies, reducing the anchoring strength by modifying the alignment layer can provide a wider sensing range [1,19].

Other Researchers have experimented on a variety of liquid crystal biosensor platforms, including the relationship between anchoring energy, birefringence, and interfacial interactions [8,10,13,18].

We aim to investigate the effect of the homeotropic anchoring energy and surface viscosity at the liquid crystal-aqueous interface on liquid crystal configuration and light transmission. In our study, we analyzed the liquid crystal bulk and interface as a function of the liquid crystal director field.

To our knowledge, mathematical modeling and computer simulation have not been extensively investigated for liquid crystal biosensors. We used a nonlinear continuum mechanics concept to analyze the deformation of liquid crystal by applying torque balance through the system. Our study used a transient two-dimensional mathematical model based on the torque balance incorporating Frank’s free energy during liquid crystal biosensor usage. The realignment of liquid crystals at the aqueous interface was also modeled with Euler–Lagrange equations.

We first characterized the interface with two dimensionless surface parameters that involve the reorientation of the whole system to analyze the sensitivity of biosensors. We simulated the confined bulk orientation of liquid crystal by manipulating the surface anchoring energy, easy axis, and interfacial properties. Secondly, we investigated the bulk orientational response due to the anchoring energy variations at the interface. We assumed that the easy axis changed rapidly from planar to homeotropic in the presence of sufficient surfactant concentration, considering that less than critical concentration can result in random orientation [15,20,21].

2. Modeling and Computational Method

2.1. Theory

In the nonlinear continuum mechanical theory of liquid crystals, the director field n defines the average orientation of the rod like molecules. When an external force is applied to the liquid crystal, the orientation will change to a new stable configuration. When the force is released, the orientation will be returning to its original state.

The total distortion free energy per unit volume of nematic liquid crystal can be calculated using the Frank elastic free energy [12,22]:

$$F_{\mathrm{d}}=\frac{1}{2}{K}_{11}{\left(\nabla ·n\right)}^2+\frac{1}{2}{K}_{22}{\left(n·\left(\nabla \times n\right)\right)}^2+\frac{1}{2}{K}_{33}{\left|n\times \left(\nabla \times n\right)\right|}^2$$

(1)

where K₁₁, K₂₂, and K₃₃ are the Frank elastic constants associated with three types of deformation, which are splay, twist, and bend, respectively.

When a system undergoes a reorientation or disturbance from the equilibrium state, the elastic constants of liquid crystals determine the restoring torques. These constants are a measure of the liquid crystal stiffness to any distortion in the system. Typically, these constants are around 10⁻¹¹ N.

A torque balance through the system is used to determine the equilibrium bulk reorientation dynamics of the liquid crystal director field in an isothermal, incompressible fluid based on Leslie-Ericksen theory. The torque balance equation in Cartesian coordinates is written in vector form as below:

$${\Gamma }_{\mathrm{e}}+{\Gamma }_{\mathrm{v}}=0$$

(2)

where Γ_v is viscous torque and Γ_e is elastic torque. The elastic torque can be calculated by using Equation (3) below:

$${\Gamma }_{\mathrm{e}}=n\times h$$

(3a)

In this equation, h is the molecular field in an equilibrium condition and is defined with its constitutive equations as below:

$$h=h_{\mathrm{S}}+h_{\mathrm{T}}+h_{\mathrm{B}}$$

(3b)

$$h_{\mathrm{S}}=K_{11}\nabla \left(\nabla ·n\right)$$

(3c)

$$h_{\mathrm{T}}=-K_{22}\left\{a\nabla \times n+\nabla \times \left(an\right)\right\}$$

(3d)

$$h_{\mathrm{B}}=K_{33}\left\{b\times \nabla \times n+\nabla \times \left(n\times b\right)\right\}$$

(3e)

where a and b are defined with the following constitutive equations:

$$a=n·\nabla \times n$$

(3f)

$$b=n\times \nabla \times n$$

(3g)

The viscous torque can be calculated by using Equation (4a)

$${\Gamma }_v=-n\times \left({\gamma }_{1}N+{\gamma }_{2}A·n\right)$$

(4a)

where ${\gamma }_1$ and ${\gamma }_2$ are the rotational and irrotational torque coefficients. The rate of strain tensor A and angular velocity tensor Ω are defined as follows:

$$A=\frac{1}{2}\left[{\left(\nabla V\right)}^{\mathrm{T}}+\nabla V\right]$$

(4b)

$$\Omega =\frac{1}{2}\left[{\left(\nabla V\right)}^{\mathrm{T}}-\nabla V\right]$$

(4c)

The system is modeled as being stationary with zero velocity since the liquid crystal in the biosensor film has no bulk movement. As a result, A and Ω are zero. Furthermore, vector N is defined as the angular velocity of the director relative to that fluid and it is expressed as follows:

$$N=\dot{n}-\Omega ·n$$

(5)

where $\dot{n}$ represents the material time derivative.

2.2. Mathematical Model

The director field n is defined in the Cartesian coordinate system, as shown in Figure 2, and described as follows:

$$n=\left(\sin \theta \cos \phi , \sin \theta \sin \phi , \cos \theta \right)$$

(6)

where θ (zenithal angle) is the polar angle between the liquid crystal director and z-axis and φ (azimuthal angle) is the angle between the projection of n on the xy plane and the x-axis as shown in Figure 2.

The polar and azimuthal angles are a function of both space and time. In the case of the two-dimensional system, however, θ and φ are functions of x, y, and time, and it is assumed that the film is infinitely wide, therefore, $\frac{\partial (\dots )}{\partial y}=0$.

The following Frank elastic free energy in two-dimensional Cartesian coordinates is obtained by substituting Equation (6) into Equation (1):

$$\begin{array}{cc}\hfill F_d=\frac{1}{2}{K}_{11}\left({sin}^2\theta \right.& {\left(\frac{\partial \phi }{\partial x}\right)}^2{sin}^2\phi +2\frac{\partial \theta }{\partial z}{sin}^2\theta \frac{\partial \phi }{\partial x}sin\phi +{\left(\frac{\partial \theta }{\partial x}\right)}^2{cos}^2\theta {cos}^2\phi -2\frac{\partial \theta }{\partial z}\frac{\partial \theta }{\partial x}sin\theta cos\theta cos\phi \hfill \\ \hfill & \left.-2\frac{\partial \theta }{\partial x}sin\theta cos\theta \frac{\partial \phi }{\partial x}sin\phi cos\phi +{\left(\frac{\partial \theta }{\partial z}\right)}^2{sin}^2\theta \right)\hfill \\ \hfill & +\frac{1}{2}{K}_{22}\left({sin}^4\theta {\left(\frac{\partial \phi }{\partial z}\right)}^2-2\frac{\partial \theta }{\partial x}{sin}^2\theta \frac{\partial \phi }{\partial z}sin\phi +{\left(\frac{\partial \theta }{\partial x}\right)}^2{sin}^2\phi +{sin}^2\theta {cos}^2\theta {\left(\frac{\partial \phi }{\partial x}\right)}^2{cos}^2\phi \right.\hfill \\ \hfill & \left.-2{sin}^3\theta cos\theta \frac{\partial \phi }{\partial x}\frac{\partial \phi }{\partial z}cos\phi +2\frac{\partial \theta }{\partial x}sin\theta cos\theta \frac{\partial \phi }{\partial x}sin\phi cos\phi \right)\hfill \\ \hfill & +\frac{1}{2}{K}_{33}\left({sin}^4\theta {\left(\frac{\partial \phi }{\partial x}\right)}^2{cos}^2\phi +{\left(\frac{\partial \theta }{\partial x}\right)}^2{sin}^2\theta {cos}^2\phi +{sin}^2\theta {cos}^2\theta {\left(\frac{\partial \phi }{\partial z}\right)}^2\right.\hfill \\ \hfill & \left.+sin\left(2\theta \right){sin}^2\theta \frac{\partial \phi }{\partial x}\frac{\partial \phi }{\partial z}cos\phi +\frac{\partial \theta }{\partial x}\frac{\partial \theta }{\partial z}sin\left(2\theta \right)cos\phi +{\left(\frac{\partial \theta }{\partial z}\right)}^2{cos}^2\theta \right)\hfill \end{array}$$

(7)

At equilibrium, in the absence of external forces, the surface energy will be minimized if the director is aligned with easy axis at the surface. Easy axis is defined as the preferred direction of the director at the surface.

The molecular orientation at the interface is characterized by two parameters: the average angle and surface anchoring. The anchoring energy is a function of two surface anchoring parameters and the polar and azimuthal deviation of liquid crystal director alignment (θ_s, φ_s) from the easy axis (θ_e, φ_e) [23].

Anchoring energy is an approximation of the energy involved in liquid crystal deformation due to surface interaction. It is determined by the Rapini-Papoular expression, which associates with the deviations of the polar and azimuthal angles from the surface easy axis [23]:

$$W\left({\theta }_{\mathrm{s}}\right)=\frac{1}{2}{W}_{\mathsf{\theta }}{\sin }^2({\theta }_{\mathrm{s}}-{\theta }_{\mathrm{e}})$$

(8a)

$$W\left({\phi }_{\mathrm{s}}\right)=\frac{1}{2}{W}_{\phi }{\sin }^2({\phi }_{\mathrm{s}}-{\phi }_{\mathrm{e}})$$

(8b)

The subscript s for θ and φ indicate the surface direction, and the subscript e indicates the easy axis or the preferred alignment at the surface. W_θ and W_φ are the polar and azimuthal anchoring parameters that describe the strength of orientation at the surface and characterize the surface anchoring. The anchoring strength, or anchoring energy coefficient, is a measure of how easily the orientation can deviate from the preferred anchoring direction.

When there is no external field, the equilibrium of a liquid crystal biosensor system is achieved by minimizing the total free energy density of the system, which includes bulk and surface energy. In a thin film with the liquid crystal-aqueous interface, the total energy of liquid crystal contained in the film can be expressed as follows:

$$F=\underset{\mathrm{V}}{{\displaystyle \int }}{F}_{\mathrm{d}}\mathrm{d}V+\underset{\mathrm{S}}{{\displaystyle \int }}{F}_{\mathrm{s}}\mathrm{d}s$$

(9)

Thus, F_d can be calculated from Equation (7), and F_s can be determined using Rapini-Papoular expression given in Equation (8):

$$F_s=\frac{1}{2}{W}_{\mathsf{\theta }}{\sin }^2({\theta }_{\mathrm{s}}-{\theta }_{\mathrm{e}})+\frac{1}{2}{W}_{\mathsf{\phi }}{\sin }^2({\phi }_s-{\phi }_{\mathrm{e}})$$

(10)

When strong anchoring is at the surface, the director aligns with the easy axis. Weak anchoring, however, allows the system to evolve towards an intermediate equilibrium state by obtaining the balance between the viscous and elastic torques depending on the anchoring strength.

The Euler–Lagrange equation can interpret the balance between generalized and frictional forces to describe the system’s distortion. This equation is derived using the calculus of variations from balancing the elastic and viscous forces at the surface to study the dynamics of thermotropic liquid crystal at the interface with the aqueous phase with the system of two generalized coordinates, θ, and φ. The Rayleigh generalized dissipation function can be simplified to drive the Euler–Lagrange equations for our system as shown below [24]:

$$\frac{\partial R^s}{\partial \dot{\varphi }}+{\Phi }_{\varnothing }=0$$

(11)

where R_s is the Rayleigh dissipation function for surfaces, ${\Phi }_{\varnothing }$ is denoted elastic forces and ɸ is represents two generalized coordinates. Then, the Euler–Lagrange equation can be written as follows:

$$\frac{\partial R}{\partial \dot{\theta }}={\mathsf{\lambda }}^s\frac{\partial \theta }{\partial t}$$

(12a)

$$\frac{\partial R}{\partial \dot{\phi }}={\lambda }^s{\sin }^2\theta \frac{\partial \phi }{\partial t}$$

(12b)

where λ^s is the surface viscosity in the liquid crystal-aqueous interface and an overdot on the function indicates time differentiation.

The contribution of surface realignment and bulk to the variation of the total free energy of the system can be written by considering Equations (7), (9) and (10) as below:

$$F=\underset{V}{{\displaystyle \int }}{F}_d\left(\varphi , \nabla \varphi \right)dV+\underset{S}{{\displaystyle \int }}{F}_s\left(\varphi \right)ds$$

(13)

The variation of F can be written as below:

$$\delta F=\underset{V}{{\displaystyle \int }}\left(\frac{\partial F_d}{\partial \varphi }\delta \varphi +\frac{\partial F_d}{\partial \nabla \varphi }\delta \nabla \varphi \right)dV+\underset{S}{{\displaystyle \int }}\left(\frac{\partial F_S}{\partial \varphi }\delta \varphi \right)dS$$

(14)

By applying the divergence theorem, the equation is as follows:

$$\delta F=\underset{V}{{\displaystyle \int }}\left(\frac{\partial F_d}{\partial \varphi }\delta \varphi -\nabla \frac{\partial F_d}{\partial \nabla \varphi }\delta \nabla \varphi \right)dV+\underset{S}{{\displaystyle \int }}\left(\frac{\partial F_d}{\partial \varphi }\delta \varphi ·\mathbf{\nu }+\frac{\partial F_S}{\partial \varphi }\delta \varphi \right)dS$$

(15)

In this equation, ν is the unit normal vector directed outward from the enclosing surface S of volume V. Then, the elastic forces at the surface can be expressed as follows:

$${\mathsf{\Phi }}_{\varphi }=\frac{\partial F_{\mathrm{d}}}{\partial \nabla \varphi }\mathbf{\nu }+\frac{\partial F_{\mathrm{s}}}{\partial \varphi }$$

(16)

2.3. Governing Equation and Auxiliary Conditions

In order to nondimensionalize the governing equations and boundary conditions, the following scaling relations are used:

$${\eta }_i{}^{*}=\frac{{\eta }_i}{{\gamma }_1},\hspace{1em}\left(\mathrm{for} i=1, 2, 3\right)$$

(17)

$$K_{ii}{}^{*}=\frac{K_{ii}}{K_{22}},\hspace{1em}\left(\mathrm{for} i=1, 2, 3\right)$$

(18)

$$x^{*}=\frac{x}{L}$$

(19)

$$z^{*}=\frac{z}{h}$$

(20)

$$t^{*}=\frac{t{K}_{22}}{{\gamma }_1{h}^2}$$

(21)

In these relations, superscript asterisks denote dimensionless variables. where L and h are the length along x and the thickness of the film along the z axis. The nondimensional parameters of surface viscosity and anchoring energy are defined as follows:

$$W=\frac{1}{2}\frac{h{W}_o}{K_{22}}$$

(22)

$${\lambda }^s{}^{*}=\frac{2{\lambda }^s}{{\gamma }_{1}h}$$

(23)

The governing equations consist of a set of two coupled transient second order partial differential equations obtained by incorporating Equations (2)–(6), with the scaling relation, Equations (17)–(23), they are as follows:

$$\begin{array}{cc}\hfill {\eta }_1^{*}\frac{\partial \theta }{\partial t^{*}}={\kappa }_1\frac{{\partial }^2\theta }{\partial x^{*2}}& +{\kappa }_2\frac{{\partial }^2\phi }{\partial x^{*2}}+{\kappa }_3\frac{{\partial }^2\theta }{\partial x^{*}\partial z^{*}}+{\kappa }_4\frac{{\partial }^2\phi }{\partial x^{*}\partial z^{*}}+{\kappa }_5\frac{{\partial }^2\theta }{\partial z^{*2}}+{\kappa }_6\frac{{\partial }^2\phi }{\partial z^{*2}}+{\kappa }_7{\left(\frac{\partial \theta }{\partial x^{*}}\right)}^2+{\kappa }_8{\left(\frac{\partial \theta }{\partial z^{*}}\right)}^2+{\kappa }_9{\left(\frac{\partial \phi }{\partial x^{*}}\right)}^2\hfill \\ \hfill & +{\kappa }_{10}{\left(\frac{\partial \phi }{\partial z^{*}}\right)}^2+{\kappa }_{11}\frac{\partial \theta }{\partial z^{*}}\frac{\partial \phi }{\partial z^{*}}+{\kappa }_{12}\frac{\partial \theta }{\partial x^{*}}\frac{\partial \theta }{\partial z^{*}}+{\kappa }_{13}\frac{\partial \theta }{\partial x^{*}}\frac{\partial \phi }{\partial z^{*}}+{\kappa }_{14}\frac{\partial \theta }{\partial z^{*}}\frac{\partial \phi }{\partial x^{*}}+{\kappa }_{15}\frac{\partial \phi }{\partial x^{*}}\frac{\partial \phi }{\partial z^{*}}\hfill \\ \hfill & +{\kappa }_{16}\frac{\partial \theta }{\partial x^{*}}\frac{\partial \phi }{\partial x^{*}}-{\eta }_2^{*}\frac{\partial \phi }{\partial t^{*}}{}^{*}\hfill \end{array}$$

(24)

$$\begin{array}{c}{\eta }_3^{*}\frac{\partial \phi }{\partial t^{*}}={\kappa }_{17}\frac{{\partial }^2\theta }{\partial x^{*2}}+{\kappa }_{18}\frac{{\partial }^2\phi }{\partial x^{*2}}+{\kappa }_{19}\frac{{\partial }^2\theta }{\partial x^{*}\partial z^{*}}+{\kappa }_{20}\frac{{\partial }^2\phi }{\partial x^{*}\partial z^{*}}+{\kappa }_{21}\frac{{\partial }^2\phi }{\partial z^{*2}}+{\kappa }_{22}{\left(\frac{\partial \theta }{\partial x^{*}}\right)}^2+{\kappa }_{23}{\left(\frac{\partial \phi }{\partial x^{*}}\right)}^2+{\kappa }_{24}\frac{\partial \theta }{\partial x^{*}}\frac{\partial \theta }{\partial z^{*}}\\ +{\kappa }_{25}\frac{\partial \theta }{\partial x^{*}}\frac{\partial \phi }{\partial z^{*}}+{\kappa }_{26}\frac{\partial \theta }{\partial z^{*}}\frac{\partial \phi }{\partial x^{*}}+{\kappa }_{27}\frac{\partial \phi }{\partial x^{*}}\frac{\partial \phi }{\partial z^{*}}+{\kappa }_{28}\frac{\partial \theta }{\partial x^{*}}\frac{\partial \phi }{\partial x^{*}}+{\kappa }_{29}\frac{\partial \theta }{\partial z^{*}}\frac{\partial \phi }{\partial z^{*}}\end{array}$$

(25)

where the spatially and angle-dependent elastic functions {κ_i}, i = 1, 2, …, 29 and viscosity functions {η_i}, i = 1, 2, 3 are provided in the Appendix A.

By using Equations (7)–(16) and applying the scaling relation, Equations (17)–(23), the following first order transient partial differential equations are obtained to simulate liquid crystal motion at the aqueous interface [24,25]. The governing equation must also satisfy this nonlinear transient boundary condition. Modeling the motion of liquid crystals at the interface is based on the Euler–Lagrange equation as follows:

$${\kappa }_{30}\frac{\partial \theta }{\partial z^{*}}=-{\lambda }^{\mathrm{s}*}\frac{\partial \theta }{\partial t^{*}}-{\kappa }_{31}\frac{\partial \phi }{\partial x^{*}}-{\kappa }_{32}\frac{\partial \theta }{\partial x^{*}}-W_{\mathsf{\theta }}\sin \left[2\left(\theta -{\theta }_{\mathrm{e}}\right)\right]$$

(26)

$${\kappa }_{33}\frac{\partial \phi }{\partial z^{*}}=-{\lambda }^{\mathrm{s}*}{\sin }^2\theta \frac{\partial \phi }{\partial t^{*}}-{\kappa }_{34}\frac{\partial \phi }{\partial x^{*}}-{\kappa }_{35}\frac{\partial \theta }{\partial x^{*}}-W_{\mathsf{\phi }}\sin \left[2\left(\phi -{\phi }_{\mathrm{e}}\right)\right]$$

(27)

where the angle-dependent elastic functions {κ_i}, i = 30, 31, …, 35 are given in the Appendix A.

The initial condition is obtained by simulating the confined liquid crystal between the Dirichlet boundary conditions and assuming that the glass surface at the bottom of the film is functionalized to create strong homeotropic anchoring in the absence of any external fields [9]. In order to model the dynamics of liquid crystal interface alignment, the top surfaces of the film are defined with Equations (26) and (27). The dimensionless initial and boundary conditions are defined as follows:

$$\theta ={\theta }_{\mathrm{i}}\left(x^{*},z^{*},t^{*}\right) \mathrm{at} t^{*}=0, 0\le x^{*}\le 3.0, 0 \le z^{*}\le 1.0$$

(28a)

$$\phi ={\phi }_{\mathrm{i}}\left(x^{*},z^{*},t^{*}\right) \mathrm{at} t^{*}=0, 0\le x^{*}\le 3.0, 0 \le z^{*}\le 1.0$$

(28b)

$$\theta =0 rad \mathrm{at} t^{*}=0,0\le x^{*}\le 3.0, z^{*}=0$$

(28c)

$$\phi =\frac{1}{4}\pi rad \mathrm{at} t^{*}=0, 0\le x^{*}\le 3.0, z^{*}=0$$

(28d)

$$\theta =\frac{1}{2}\pi rad \mathrm{at} t^{*}=0, x^{*}=0 , 0\le z^{*}<1.0$$

(28e)

$$\phi =\frac{1}{4}\pi rad \mathrm{at} t^{*}=0, x^{*}=0 , 0\le z^{*}<1.0$$

(28f)

$$\theta =-\frac{1}{2}\pi \mathrm{rad} \mathrm{at} t^{*}=0, x^{*}=3.0 , 0\le z^{*}<1.0$$

(28g)

$$\phi =-\frac{1}{4}\pi \mathrm{rad} \mathrm{at} t^{*}=0, x^{*}=3.0 , 0\le z^{*}<1.0$$

(28h)

where the confined film geometry is defined at $0\le x^{*}\le 3.0$ and $0\le z^{*}<1.0$ in Cartesian coordinate.

2.4. Optical Interference

The alignments of the liquid crystal director field can be observed and interpreted when white light passes through a thin film between a cross polarizer and analyzer [8,18]. The anisotropy of liquid crystal results in splitting the incoming white light into two incident rays, which vibrate orthogonally to each other. Therefore, depending on the relation between the polarization light and the director, there will be two different refractive indexes. It is called the ordinary refractive index n_o when the light is perpendicular to the director. When the light is parallel to the optical axis, it is referred to as an extraordinary refractive index n_e [18].

Thus, the phenomenon of birefringence occurs in liquid crystal because of the difference between two refractive indices. As mentioned previously, the reorientation of the liquid crystal changes its optical properties. During the system deformation, when the light propagates through the film, it experiences an effective extraordinary refractive index. The ordinary refractive index is constant and independent of light propagation; however, the extraordinary refractive index varies depending on the angle between the beam of light and the director.

Hence, the effective refractive index n_eff and effective birefringence Δn_eff will be defined as follows [26,27]:

$$n_{\mathrm{eff}}=\frac{n_{\mathrm{o}}{n}_{\mathrm{e}}}{\sqrt{n_{\mathrm{e}}^2{\cos }^2\theta +n_{\mathrm{o}}^2{\sin }^2\theta }}$$

(29)

$$\mathsf{\Delta }{n}_{\mathrm{eff}}=n_{\mathrm{eff}}-n_{\mathrm{o}}$$

(30)

where θ is the angle between the direction of propagation light and the optical axis.

By substituting Equation (29) into Equation (30) and integrating through its thickness of the thin film, effective birefringence can be calculated along z-direction as follows:

$$\mathsf{\Delta }{n}_{\mathrm{eff}}={{\displaystyle \int }}_0^d\left(\frac{n_{\mathrm{o}}{n}_{\mathrm{e}}}{\sqrt{n_{\mathrm{e}}^2{\cos }^2\theta +n_{\mathrm{o}}^2{\sin }^2\theta }}-n_{\mathrm{o}} \right)\mathrm{d}z$$

(31)

The relative phase lag (δ) is also referred to as the optical retardance introduced when light propagates through a film, and it is defined as below [18]:

$$\delta ={{\displaystyle \int }}_0^d\frac{2\pi }{\lambda }\mathsf{\Delta }{n}_{\mathrm{eff}}\mathrm{d}z$$

(32)

where λ is the wavelength of the light, d is the distance between the two cross polarizers.

Since the incident of linearly polarized light is elliptically polarized as it passes through the birefringence medium between two cross polarizers; therefore, the relative transmitted intensity of the interference in the z-direction of the film is calculated as follows [27]:

$$I_{\mathrm{r}}=\frac{I}{I_o}={{\displaystyle \int }}_0^d\frac{1}{2}{\sin }^2\left(2\phi \right){\sin }^2\left(\frac{1}{2}\delta \right)\mathrm{d}z$$

(33)

where φ is the angle between the analyzer and the projection of the optic axis on the xy plane. I_o is the amplitude of incoming linearly polarized light which depends on φ. According to Equation (33), the film appears dark when the optical axis aligns with one of the polarizers or analyzer directions, while the maximum light intensity occurs when the φ = 45°.

When the liquid crystal orientation is homeotropic, its optical retardation and extraordinary refractive index are zero. In this case, the propagating light does not exhibit any birefringence, and it travels along the optic axis. Therefore, regardless of the value of φ, the intensity becomes zero, and the film appears dark. Additionally, when the orientation is uniform planar and θ = 90°, the intensity will be dependent on the first term of Equation (33). Hence, the interference bright and dark intensity patterns can be observed beyond the analyzer. The grayscale mappings of relative light intensity interference can be interpreted as the result of variations in transmitted light intensity during the evolution of the system [18].

2.5. Computational Method

As part of our mathematical model development, we used Mathematica [28] software and derived all the mathematic equations. As a result, our mathematical model includes a set of nonlinear unsteady partial differential equations with Dirichlet and time-dependent boundary conditions. In this work, a system of second-order partial differential equations is solved in two dimensions using the Galerkin finite element method with linear basis function. Mesh refinement was performed in order to ensure that mesh size and time step are set appropriately. The mesh was generated with rectangular elements consisting of 300 × 100 elements to discretize the two-dimensional domain in film geometry [29,30]. In addition, the initial dimensionless time step is considered 10⁻⁴. The numerical techniques and divergence theorem are used to obtain the Galerkin form of partial differential equations and develop the residual vector and Jacobian matrix in finite element method [31].

The Newton-Raphson method is used to solve the set of nonlinear equations after spatial discretization using Galerkin finite elements. We used the implicit first-order Euler predictor controller as a time integrator. In addition, a time step controller is implemented in the program using the Gear method to optimize computation time [32,33,34]. Approximation to a solution is achieved when the difference between the two sequence solution vectors reaches 10⁻⁶.

Using FORTRAN 95 as the programming language [35], we developed the finite element program and compiled it using the Intel compiler. Next, we used the LAPACK solver routines from the Intel Math Kernel Library to solve the set of linear algebraic equations [36]. As a part of the post-processing step, we used MATLAB, SigmaPlot, and open-source Para View software to analyze and visualize the results [37]. We performed the computational analysis using high-performance computing resources at Digital Research Alliance of Canada’s Cedar and Graham clusters.

3. Result and Discussion

This section discusses the initial conditions and parametric studies, which focus on the effect of dimensionless surface viscosity λ^s* and dimensionless homeotropic anchoring W on the dynamics of biosensor film performance. We also present our numerical results of optical signals to characterize the system sensitivity.

We examined a wide range of surface viscosity parameters and anchoring parameters. The surface viscosity parameter determines the extent to which the surface rotates relative to the bulk rotational viscosity. In addition, the anchoring parameter defines how the surface anchoring strength relates to the bulk elasticity. The anchoring parameter takes into account the film thickness, anchoring strength, and liquid crystal elasticity as we defined in Equations (22) and (23).

The experiments shows that the surface anchoring and easy axis change when the surfactant concentration increases and the entire system, including the interface and bulk, is reoriented to reach the equilibrium state [13]. It has been reported that insufficient surfactant caused the tilted alignment at the interface, but with increased surfactant concentration, the homeotropic alignment was induced by the adsorption of the surfactant. However, according to the Rey model [15] and other experimental studies, higher concentrations of surfactant causes interface instability and random alignment surface alignment after exceeding the critical concentration of surfactants instead of homeotropic alignment [13,16].

Furthermore, studies have shown that anchoring energy and easy axis at the aqueous interface depend on surfactant concentrations. Yesil et al. indicated that homeotropic anchoring at the interface increases when surfactant concentrations increase in aqueous solutions within the concentration range necessary to shift from planar to homeotropic alignment [16].

Hence, we studied the reorientation of bulk due to a change in homeotropic anchoring energy. We assumed in this study that the easy axis changes abruptly in response to homeotropic anchoring at the aqueous-liquid crystal interface in the presence of sufficient surfactants when the liquid crystal aqueous interface has stability with the preferred orientation of homeotropic alignment.

Furthermore, we assumed an abrupt change of easy axis to study how the whole system gets evolved. The easy axis abruptly changes from θ = 90° to θ = 0° when sufficient surfactant concentration is present [10]. We did not consider the discontinuous anchoring transition reported by the Rey model. Instead, we assumed the concentrations have reached near the saturated concentrations and below the critical bulk concentrations and easy axis changes to a homeotropic orientation. We then analyzed how the whole system has evolved and reoriented to reach equilibrium. We used the Euler–Lagrange equation and Rayleigh dissipation function as part of this model. We assumed that the top surface of liquid crystal biosensors has the anchoring energy of homeotropic anchoring.

In this simulation, to study the temporal changes of dependent variables, the evolution of the system is performed by considering the various range of values for W and λ^s*. The surface viscosity parameter was changed from 10 to 10^4, and the anchoring energy parameter effect was studied from a range of 1 to 10⁵ for each surface viscosity case.

In all study cases, initial conditions were the hybrid initial conditions, which implied that the confined system was reoriented due to film boundary conditions. As a result, the energy level in the system was reduced prior to introducing the surfactant at the aqueous interface.

Figure 3 shows the initial liquid crystal orientation obtained due to the strong anchoring at the sides and bottom of a liquid crystal biosensor film. The bottom surface of the biosensor film is coated including glass with homeotropic anchoring alignment, and the sides of the film are gold or copper, which also induces a perpendicular anchoring alignment at the side surfaces [8].

Figure 4 shows the light intensity interference calculated through the film’s thickness representing the bulk orientation at initial conditions. As shown in the figure, the effect of the grid surface shows brightness close to the sides as reported in the experiment [18]. The alignment of the liquid crystal near the side surface is affected by the surface material. As a result, the liquid crystal inside the pores is affected by the opposite boundary conditions of two gold/copper surfaces, leading to the symmetric breaking at the middle of the film. As shown in the figure, there are symmetric breakdowns in half of the film. The pattern is consistent with experimental results as we did not include the defect in the model for modeling simplicity, and the symmetric break happened in the half of the film [1,8].

In this study, the director’s field orientation is examined as a function of time. We assumed the field variation in the y-direction is negligible. Hence, we studied the field variation only in the xz plane. In addition, the sensitivity of the field due to the surface anchoring strength parameter relative to the bulk elasticity was examined.

We studied the reorientation of the whole system of liquid crystals due to changing anchoring strengths at the interface. An increasing anchoring strength is referred to as increasing the homeotropic anchoring. In our study, we considered the easy axis as homeotropic alignment, and as mentioned earlier, we assumed an abrupt change of the easy axis due to the presence of surfactants.

Figure 5 illustrates three case studies that show the bulk reorientation at a steady state which is a result of adding the surfactant at the liquid crystal-aqueous interface that causes the director gradient to appear at the surface. Then, the bulk director starts to reorient since surface energy depends on the relation between the director at the surface and the easy axis. If the anchoring energy increases (anchoring coefficient), the surface director tends to align with the direction of the easy axis. The easy axis direction at the liquid crystal aqueous interface depends on the bounding surface and intermolecular interaction with the surfactant. As mentioned earlier, the homeotropic anchoring strength depends on the concentration of surfactants at the interface [20,38].

Figure 5c shows that at the equilibrium, surface energy will be minimized if the director is aligned with the preferred direction at the surface in the absence of external torque. Therefore, the orientation of the director is the same as the easy axis at the interface [23].

The system response for each parameter variation is obtained in the form of the optical signal. Since the film has a constant thickness, the birefringence provides information about the surface alignment. Figure 6 shows the optical light intensity using the grayscale result in the bulk reorientation in Figure 5.

The spatial and temporal tilt angle is measured relative to the surface normal following Figure 2, in the Cartesian coordinate system between 0° to 90°. The azimuthal angle on the xy plane is measured between 0° and 90° degrees from the x-axis to the projection of the director. In this model, the inner sides of grids’ pore surfaces are opposite in terms of easy axis and have a parallel easy axis direction toward the z-axis [38]. The homeotropic anchoring at the side surface of the film has been assumed that have strong anchoring; therefore, we considered the fixed boundary condition at the left and the right side of the film. In this study, the fixed boundary condition is considered as follows θ = ±90° and θ = ±45°.

Since intensity and birefringence are calculated by integrating the z-direction, the calculated value varies based on the x-axis location. Since the experimental results on biosensors deal with average values of birefringence and intensity [18], we defined the average dynamic of the orientation of the system to better characterize the performance dynamics. Using the following formula, the mean magnitude of the orientation of the polar angle $\langle \parallel \theta \parallel \rangle$ can be calculated:

$$\langle \parallel \theta \parallel \rangle =\frac{1}{l\times h}{\displaystyle \iint }\theta \mathrm{d}x\mathrm{d}z$$

(34)

Figure 7 shows a semilogarithmic plot of the magnitude of mean polar orientation angle at steady-state conditions <||θ_ss||> versus homeotropic anchoring strength W. It depicts the average orientation of liquid crystal bulk in the biosensor film at the steady-state condition in relation to W and λ* with the sigmoidal trend when the parametric values for W in the parametric studies are appropriate. In the range of 10² ≤ W ≤ 10⁵, the average polar angle through the film reaches the minimum value. Furthermore, the average angle is constant in this range. This suggests that surface viscosity does not affect bulk reorientation for strong homeotropic anchoring W ≥ 10² and surface viscosity λ* ≤ 10³. Therefore, the relation between average angle and W is independent of dimensionless surface viscosity in this range. Furthermore, the dark appearance of film interference at the steady state does not vary with surface viscosity within this range. However, the surface viscosity affects the transient process in the film during the biosensor performance.

In addition, within the range of 10⁰ < W < 10^2, the mean angle differs with surface viscosity. Furthermore, the mean angle decreases with increasing of homeotropic anchoring and decreasing surface viscosity. This suggests that the bulk reorientation resulting from top surface contribution is affected by anchoring energy and surface viscosity at the steady-state condition. Furthermore, by examining the higher viscosity λ* ≥ 10⁴, the W = 10³ is sufficient to obtain a complete dark appearance during the biosensor performance.

We examined the results of reorientation at x^* = 0.2, x^* = 0.75, and x^* = 1.3 and observed that the angle orientation at x^* = 0.75 was consistent with the calculated mean angle orientation during the event. Therefore, we considered analyzing the optical results at x ^* = 0.75 for our investigation. Figure 8 shows the relationship of effective birefringence with W and λ^* at x ^* = 0.75. It illustrated that the trend of reducing birefringence when W increases. Equations (26) and (27) illustrate that the tendency of the director at the surface to align with the easy axis depends on W. The average angle between the optic axis and incident light reduces to zero when the system approaches the new equilibrium state. Therefore, calculated birefringence decreases according to Equation (31).

The results demonstrate the influence of surface viscosity compared to the rotational viscosity of bulk in response to increasing homeotropic anchoring energy. As a result, higher surface viscosity results in brighter optical interference at the steady state.

Since the birefringence is reduced gradually, a relaxation characteristic time τ is defined [20]. As a result, it is calculated as follows:

$$\mathsf{\Delta }{n}_{\mathrm{eff}}={\mathrm{e}}^{-t/\tau }\mathsf{\Delta }n$$

(35)

A time constant is defined as 36.78 percent of the initial birefringence. Furthermore, we described the characteristic time as 35% of the angle orientation to investigate the temporal average orientation.

Figure 9 shows the relationship between W and τ on a logarithmic scale. The plot shows that τ decreases as W increases. τ has a reasonably linear relationship with W in the range of 10² < λ* ≤ 10³, with the constant rate dependent on the surface viscosity.

However, for λ* ≤ 10², τ decreases gradually with increasing W. For W ≤ 10³, the relation between τ and W remains linear for 10 < λ* ≤ 10³. This suggests that the bulk reorientation due to surface realignment is sensitive to surface viscosity and anchoring strength. Furthermore, at the lower surface viscosity λ* ≤ 10, in the range of W ≥ 10³, the relation between τ and W remains constant, which suggests that the reorientation of the bulk is not affected by the strong homeotropic anchoring for W ≥ 10³. Additionally, it shows that τ is almost the same at W = 10⁵, which implies that τ is independent of surface viscosity at very high homeotropic anchoring strengths.

Our optical results show that the optical interference reaches the final light intensity at 3τ when the film reaches its minimum birefringence and appears dark. In Figure 10 and Figure 11, we plot the reciprocal time at 2τ and 3τ of the process in relation to W.

Scaling the Results

Using the data analysis method [39], the effective birefringence versus time constant is scaled. Figure 12 is constructed by intersecting the two tangents on the birefringence versus time constant data. The figure below shows a plot of scaled birefringence versus scaled time constant. This plot shows how effective birefringence declines exponentially. Furthermore, the scaled plot shows that the response time of biosensor undergoes an exponential decline and slows down about 1.5 characteristic times.

According to the scaled data obtained from our model, all study cases follow the same physics as shown in Figure 12. However, we were not able to find a similar experimental study to investigate the reason for this behavior. Further experimental study is required.

4. Summary and Conclusions

Our study is on a model for simulating liquid crystal transient alignment at the interface of a film using torque balance, and the Euler–Lagrange equation was developed and implemented. As a result, the optical texture was obtained and discussed. Furthermore, our model estimates the performance regimes of a liquid crystal biosensor system recognizable by its optical signal. Using the scaling parameters, we can predict the system’s sensitivity and temporal dependent variables to identify the system’s tendency for reorientation following changes in the film’s top surface. Hence, we demonstrated the sensitivity of a liquid crystal film biosensor to changes in surface viscosity, homeotropic surface anchoring strength, bulk elasticity, and bulk rotational viscosity. The simulation result was consistent with previous experiments’ observations, and the model can predict the birefringence and interference patterns throughout time and system reorientation.

Additionally, our model assumption in rapid changes in the easy axis was validated by comparing the simulation results with experimental data. As a result of our ability to predict and control the behavior of liquid crystals, we will have the ability to optimize the performance of the film in conjunction with experimental data at each specific biomolecular interface. To improve the accuracy of the calculation, we recommend considering the thermodynamics of the interface to define the easy axis due to the surfactant’s influence factor.