Colloquium: Phase transitions in polymers and liquids in electric fields

The structure and thermodynamic state of a system changes under the influence of external electric fields. Neutral systems are characterized by their dielectric constant $\epsilon$, while charged ones also by their charge distribution. In this Colloquium several phenomena occurring in soft-matter systems in spatially uniform and nonuniform fields are surveyed and the role of the conductivity $\sigma$ and the linear or nonlinear dependency of $\epsilon$ on composition are identified. Uniform electric fields are responsible for elongation of droplets, for destabilization of interfaces between two liquids, and for mixing effects in liquid mixtures. Electric fields, when acting on phases with mesoscopic order, also give rise to block-copolymer orientation, to destabilization of polymer-polymer interfaces, and to order-order phase transitions. The role of linear and nonlinear dependences of $\epsilon$ on composition will be elucidated in these systems. In addition to the dielectric anisotropy, existence of a finite conductivity leads to appearance of large stresses when these systems are subject to external fields and usually to a reduction in the voltages required for the instabilities or phase transitions to occur. Finally, phase transitions which occur in nonuniform fields are described and emphasis on the importance of $\epsilon$ and $\sigma$ is given.

Figure 1

(Color online) Two liquids in uniform electric field. (a) Schematic of two liquids with permittivities ${\epsilon }_1$ and ${\epsilon }_2$ confined between two plates at separation $L$ and voltage difference $V$. The pressure on the interface located at $y=h$ is given by Eq. (9). (b) An initially flat interface at $h=L∕2$ can break into two parts. The electrostatic energy, given by Eq. (10), prefers $h=0$ or $h=L$. The preferred state is shown in (c).

Figure 2

(Color online) Stabilization and destabilization by electric field. (a) Two liquids in perpendicular electric field. $h_0$ is the unperturbed interface location. Interface perturbations are unstable if their wavelength is long enough. (b) A liquid film is stabilized in tangential electric field.

Figure 3

The most unstable wavelength ${\lambda }^{*}$ vs reduced electric field for various thin films. $E_1$ and $E_2$ are the electric fields in the two liquids, ${\lambda }_0=V^2{\left(\Delta \epsilon \right)}^2∕\gamma {\left({\epsilon }_1{\epsilon }_2\right)}^{1∕2}$, and $\tilde{E}=V∕{\lambda }_0$. The expected relation from Eq. (16), ${\lambda }^{*}∕{\lambda }_0=2\pi {(E_1{E}_2∕{\tilde{E}}^2)}^{-3∕4}$, is the straight line with slope of $-3∕4$. Adapted from 47.

Figure 4

Pattern transfer. Use of topographically patterned electrodes enables transfer of the top electrode onto the polymer liquid, as is depicted schematically in (a). Tapping-mode atomic force microscopy images in (b) and (c) show the resulting topography of the polymer layer. Adapted from 85.

Figure 5

Dispersion relation between wave number [scaled by ${({\epsilon }_2{V}^2∕\gamma L^3)}^{1∕2}$] and inverse growth time (scaled by ${\eta }_1\gamma L^3∕{\epsilon }_2^2{V}^4$). Curves 1–3 are for pure dielectric liquids [Eq. (14)]. Curves 4–6 are from the leaky dielectric model with small dimensionless conductivity $S$. The conductivity in curve 7 is infinite. From 77.

Figure 6

Orientation of PS-PMMA block-copolymer lamellae. (a) The bulk experiments of 4 showed near-perfect alignment of lamellae in the direction parallel to the external field. (b) Time evolution of the orientational order parameter $S$ for two samples with different cooling rates as measured by birefringence experiments. Solid line is an exponential fit. Adapted from 4. (c) TEM closeup of cylindrical diblock copolymers near an electrode. Inset: Schematic of the full setup. Dashed arrows are field lines and box indicates area of image. Note how cylinders curve and follow the field lines near the corner. From 61.

Figure 7

Summary of sample orientation with respect to the field. Orientation angle is the angle between cylinders’ axis and external field for three different film thicknesses $L$. Shaded area is a coexistence “mixed” region, where both orientations are observed. Critical fields $E_1$ and $E_2$ mark the transition fields between the three possible states (cylinders parallel or perpendicular to the surface and a mixed state). Adapted from 97.

Figure 8

Inverse exponential relaxation time $\eta ∕\tau$ vs $E_0^2$ from four copolymer solutions as obtained by SAXS. When properly scaled, most data sets fall on a single universal curve. From 88.

Figure 9

(Color online) Capacitor analogy for three competing lamellar structures confined in a condenser. (a) Parallel and (b) perpendicular layerings and (c) mixed morphology, consisting of two parallel layers of thickness $a$ at the electrodes and perpendicular lamellae throughout the rest of the film. The equivalent electrical circuits are illustrated on the left.

Figure 10

(Color online) Stability diagrams for confined lamellar phases in the strong-segregation regime in electric fields. (a) $L=10d$ and is fixed. Small value of $E_0$ leads to parallel layers. Increase in $E_0$ leads to a direct transition to perpendicular layers when $\delta =({\gamma }_{\mathrm{BS}}-{\gamma }_{\mathrm{AS}})∕{\gamma }_T$ is small enough. If $\delta$ is large, $\delta >{\delta }^{*}$, increase in the field induces first a transition to a mixed phase and subsequently to a perpendicular orientation. (b) Diagram in the $E_0-L$ plane with fixed $\delta =5$. $E_0$ is scaled by ${({\gamma }_T∕d)}^{1∕2}$. Adapted from 99.

Figure 11

TEM cross section of thin-film PS-PMMA block copolymer annealed in electric field $E_0\simeq 40\mathrm{V}∕\mu \mathrm{m}$. Few lamellae lie parallel to the substrate (bottom) and polymer-air interface (top), while the rest of the film has lamellae parallel to the field. Adapted from 113.

Figure 12

Schematic of two normal-field instabilities in lamellar block copolymers (not to scale). $d$ is the unperturbed lamellar thickness and ${\lambda }^{*}$ is the undulation period. (a) In the strong-segregation regime, adjacent lamellae bend in phase with each other. ${\lambda }^{*}$ is macroscopic, ${\lambda }^{*}⪢d$. (b) Close to the ODT point, lamellae are easier to deform, and adjacent lamellae undulate out of phase with each other. ${\lambda }^{*}$ is equal to the lamellar thickness $d$.

Figure 13

(Color online) Block-copolymer phase diagrams. (a) Phase diagram for diblock copolymers in the $({\varphi }_0,\chi )$ plane at fixed electric field. Two outer solid lines are zero-field hexagonal-to-bcc and bcc-to-disorder phase boundaries. Lines A and B are transitions from distorted bcc to the hexagonal and disordered phases, respectively, and line C is the boundary between hexagonal and disordered phases in electric field given by ${({\epsilon }_{0}N{v}_0∕k_{B}T)}^{1∕2}{E}_0=0.2$. They meet at the triple point $({\varphi }_0,N\chi )=(0.39,11.43)$. Adapted from 45. (b) Phase diagram in the $(E_0,\chi )$ plane and fixed composition ${\varphi }_0=0.3$. For low values of field, a distorted bcc phase lies between the disordered (Dis) and hexagonal phases. The phase lines terminate at a triple point $({\chi }_{\mathrm{tr}},E_{\mathrm{tr}})$. When $E_0>E_{\mathrm{tr}}$, the distorted bcc does not exist. Solid lines are from analytical one-mode approximation, whereas dashed lines are from self-consistent field theory. The axes of each calculation are scaled by the relevant values of ${\chi }_{\mathrm{tr}}$ and $E_{\mathrm{tr}}$. Adapted from 45 and 101.

Figure 14

(Color online) Three model systems where field gradients lead to demixing. (a) A single charged colloid of charge $Q$ and radius $R_1$. (b) A charged wire with charge density $\lambda$ per unit length and radius $R_1$ or two concentric cylinders with radii $R_1$ and $R_2$. (c) A wedge comprised of two flat electrodes with an opening angle $\beta$ and potential difference $V$. $R_1$ and $R_2$ are the minimal and maximal values of the distance $r$ from the imaginary meeting point.

Figure 15

(Color online) Critical demixing fields. (a) Scaled critical voltage for demixing $M_w^{*}$ for a closed wedge as a function of distance from the transition (binodal) composition ${\varphi }_t$. Symbols are numerical values and line is approximate analytical expression [Eq. (42)]. (b) Scaled critical charge of spherical and cylindrical colloids as a function of temperature. Open symbols are $M_c^{*}$ for a closed cylindrical system of compositions ${\varphi }_0=0.2$ (squares), 0.3 (circles), and 0.4 (crosses). Filled symbols are $M_s^{*}$ for an isolated spherical colloid coupled to a reservoir with the same compositions. From 50.