Flow provides a strong aligning force. However, because of boundary constraints and conditions of continuity, the potential for flow-induced orientation is limited. Electric fields provide a weaker aligning force but offer the advantage of local alignment control by application of spatially specific electric fields. For these reasons, electric fields may provide a unique pathway to new applications for block copolymers. In addition, electric field alignment is scientifically interesting because the driving force for alignment is much simpler than that induced by flow. Studies of field alignment can be used to learn about materials properties like defect mobilities and give straightforward insight into alignment mechanisms.
Investigations on solvent-based systems of different block copolymers will be described in chapters 5 to 8.
1.3.2.1. Electrothermodynamics of Microphase-Separated Block Copolymers
As an example we consider a block copolymer filling the gap between parallel, planar electrodes. The electrodes are held at a constant potential by a voltage source. Therefore, the free energy of the system contains an electrostatic contribution54:
r ε(r): local dielectric constant
E(r): electric field
The integration is over the volume of the material, V.
According to this expression, materials with a high dielectric constant are attracted to regions of high field strength, in order to maximize the magnitude of the negative electrostatic contribution to the free energy. Different composition patterns within a block copolymer material and the associated spatially varying local dielectric constant produce different patterns of electric field. The consequence is a composition-pattern dependent electrostatic
contribution to the free energy. Certain orientations of the composition pattern are thermodynamically favored over others. The forces that give rise to preferred orientations arise from field-induced polarization charges resulting from gradients in the dielectric constants. Using Maxwell´s equation for dielectric materials, we find that polarization charges exist wherever the gradient in the dielectric constant has a component in the direction of the applied field.
We will now consider the two different microdomain orientations of a lamellar AB diblock copolymer parallel and perpendicular to the electrodes. For the parallel orientation, symmetry in the transverse direction dictates a vertical electric field everywhere with the continuity condition at the AB interface:
εAEA = εBEB , Equation 1-7
where εA and εB are the dielectric constants of the materials A and B, and EA and EB are the electric field strengths in regions of the materials A and B. The discontinuity in the field strength at the phase boundaries arises from excess polarization charges. The total voltage drop across the lamellae must equal the applied voltage, V:
EAφAd + EBφBd = V, Equation 1-8
Where φA and φB are the volume fractions of components A and B and d is the electrode spacing. Equations 1-7 and 1-8 yield the field strengths:
EA =
Knowing the field strength, the electrostatic contribution to the free energy can be derived from Equation 1-6:
where εh is the harmonic average of the dielectric constants:
B
This gives the effective dielectric constant of the material in the parallel orientation.
If the lamellae are perpendicular to the electrode surfaces, all the interfaces are parallel to the electric field and thus there will be no excess polarization charges. The field will be a constant V/d throughout the material. The electrostatic contribution to the free energy can be calculated as
where εa is the arithmetic average of the dielectric constants:
εa = φAεA + φBεB Equation 1-13
This is the effective dielectric constant of the material in the perpendicular orientation.
As the arithmetic mean of the dielectric constant always matches or exceeds the harmonic mean, the perpendicular orientation always represents the lower energy state:
εa - εh = (εA - εB)2
The effective dielectric constant for an arbitrary orientation where the lamellar normal vector forms a tilt angle, θ, with respect to the vertical can be expressed as follows:
εeff(θ) = εa + (εh - εa)cos2 θ Equation 1-15
The material experiences a torque whenever the lamellae are not perpendicular to the electrodes. The torque arises from the forces of attraction between the excess polarization charges at the interfaces and the charges on the electrodes.
In the above considerations, the anisotropic behavior of the block copolymer microstructure arises from shape and not from molecular anisotropy. In contrast to work by Gurovich55 - 57, contributions to the electrostatic free energy from the alignment and stretching
of chains and their difference in polarizability in the direction along and perpendicular to the bonds has been neglected.
In the following, we will briefly describe the calculation of the electrostatic contributions to the free energy, based on Equation 1-6, as derived by Amundson et al.58:
The local dielectric constant in a block copolymer sample is a function of local composition and can be expressed as an expansion in the composition pattern, ψ , associated with the ordered state:
Here, β characterizes the sensitivity of the dielectric constant to compositional change:
β = δε/δψ, εD is the dielectric constant in the limit of vanishing stationary composition pattern and includes a contribution from dynamic composition fluctuations. The effect of dynamic fluctuations is separated from the effect of the composition pattern, ψ , associated with the ordered phase. Since the dynamic fluctuations have short correlation lengths, they will not significantly couple to an electric field. The stationary composition pattern can have a much larger correlation length and can couple more effectively to an electric field.
Using Maxwell´s equation, ∇ [ε(r) E(r)] = 0, and Equation 1-6, the electrostatic contribution to the free energy density for lamellar microstructure can be written as:
( )
of the applied electric field E0. ε denotes the space-averaged dielectric constant.Only the first term in the brackets is anisotropic and contributes towards alignment. The free energy is minimized, when the wave vectors are in the plane perpendicular to E, i.e. the lamellar planes contain E. The alignment force is proportional to the square of the applied field strength E02, the mean square of the composition pattern, ψ 2 , and the material parameter β2/εD.
The size of the anisotropic component of the electric energy is rather small, i.e. for the energy difference between aligned and misaligned orientations of a region to equal kBT, the
region must be of order of some hundred nanometers for a PS-b-PMMA block copolymer.
The electric field can only affect the microdomains if it is acting on an organized state with long-range order.