Predictive modeling of nanoscale domain morphology in solution-processed organic thin films

The electronic and optoelectronic properties of molecular semiconductor thin films are directly linked to their extrinsic nanoscale structural characteristics such as domain size and spatial distributions. In films prepared by common solution-phase deposition techniques such as spin casting and solvent-based printing, morphology is governed by a complex interrelated set of thermodynamic and kinetic factors that classical models fail to adequately capture, leaving them unable to provide much insight, let alone predictive design guidance for tailoring films with specific nanostructural characteristics. Here we introduce a comprehensive treatment of solution-based film formation enabling quantitative prediction of domain formation rates, coverage, and spacing statistics based on a small number of experimentally measureable parameters. The model combines a mean-field rate equation treatment of monomer aggregation kinetics with classical nucleation theory and a supersaturation-dependent critical nucleus size to solve for the quasi-two-dimensional temporally and spatially varying monomer concentration, nucleation rate, and other properties. Excellent agreement is observed with measured nucleation densities and interdomain radial distribution functions in polycrystalline tetracene films. Numerical solutions lead to a set of general design rules enabling predictive morphological control in solution-processed molecular crystalline films.

Figure 1

(a) Schematic of the deposition apparatus. The components are housed inside a sealed nitrogen-filled chamber. TET vapor produced in a heated crucible is carried to a squalane-coated ITO-glass substrate by a stream of gas exiting through a tapered nozzle, impinging with axisymmetric stagnation point flow resulting in uniform deposition over a $1\text{−}\mathrm{c}{\mathrm{m}}^2$ area. The substrate temperature is held fixed by a thermoelectric element. Film growth is monitored in situ via fluorescence videomicroscopy through a viewport in the lid. (b) Epifluorescence micrograph showing a representative TET film. The scale bar is 100 µm. (c) Chemical structures of squalane and tetracene.

Figure 2

Time-dependent nucleation rate for TET films prepared using four different flux rates. The flux in units of ${10}^{21}$ monomers ${\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$ is indicated for each sample. Nucleation occurs in a brief burst, preceded by an induction regime and followed by a growth regime. The inset shows the time evolution of the number of crystals per unit area. Solid lines result from fitting using Eqs. (1).

Figure 3

Critical nucleus size $i^{*}$ (left axis) and the concentration of critical clusters $P(i^{*},n)$ (right axis) as functions of time, obtained from solving Eqs. (1, 2, 3) using the same parameters as in Fig. 2. The flux in units of ${10}^{21}$ monomers ${\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$ is indicated for each calculation. Note the correspondence of a minimum in critical cluster size with a maximum in the concentration of critical clusters, the driver of burst nucleation.

Figure 4

At the beginning of the nucleation regime, the first crystals to become visible are randomly positioned with nearest-neighbor (NN) spacing close to a CSR (2D Poisson) distribution (line). As the density increases, the mean NN spacing remains larger than that predicted by the 2D Poisson distribution, indicating suppressed nucleation in the vicinity of crystals that have already formed due to competition for monomers. Squares denote $F=8.0\times {10}^{21}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$; circles, $F=11.0\times {10}^{21}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$; upward triangles, $F=12.8\times {10}^{21}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$; and downward triangles, $F=18.0\times {10}^{21}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$. The inset compares the experimental (bars) and Poisson (line) NN spacing probability distributions at the end of the nucleation regime for $F=18.0\times {10}^{21}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$.

Figure 5

Comparison of the experimental crystal separation probability distribution to the predicted distribution given by the treatment described in the text (solid lines). The inset shows the unscaled distribution. Squares denote $F=8.0\times {10}^{21}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$; circles, $F=11.0\times {10}^{21}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$; upward triangles, $F=12.8\times {10}^{21}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$; and downward triangles, $F=18.0\times {10}^{21}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$. Line color is matched to symbol color.

Figure 6

Scaling behavior of $N_{\max }$ determined from numerical integration of Eqs. (1). Lines are a power-law fit to a representative series in each set, chosen to correspond to the average parameters found from experiment. For each figure part and symbol type the parameters are given as $\{\sigma \left(\mathrm{N}{\mathrm{m}}^{-1}\right),\gamma$ (unitless), $D\left({\mathrm{m}}^2{\mathrm{s}}^{-1}\right),F\left({\mathrm{m}}^{-3}{\mathrm{s}}^{-1}\right)\}$ as follows: (a) squares $\{0.017,18.0,7.58\times {10}^{-11},\text{variable}\}$, downward triangles $\{0.021,13.0,7.58\times {10}^{-11},\text{variable}\}$, upward triangles $\{0.0235,13.0,7.58\times {10}^{-11},\text{variable}\}$, circles $\{0.0205,18.0,7.58\times {10}^{-11},\text{variable}\}$, diamonds $\{0.021,18.0,7.58\times {10}^{-11},\text{variable}\}$, and solid hexagons are experimentally measured values; (b) squares $\{0.013,13.0$, variable, $1.05\times {10}^{22}\}$, downward triangles $\{0.021,10.0$, variable, $1.05\times {10}^{22}\}$, upward triangles $\{0.021,13.0$, variable, $1.05\times {10}^{22}\}$, circles $\{0.021,18.0$, variable, $1.05\times {10}^{22}\}$, and diamonds $\{0.03,13.0$, variable, $1.05\times {10}^{22}\}$; (c) squares $\{0.012$, variable, $7.58\times {10}^{-11},1.05\times {10}^{22}\}$; downward triangles $\{0.021$, variable, $7.58\times {10}^{-11},1.45\times {10}^{22}\}$, upward triangles $\{0.021$, variable, $7.58\times {10}^{-11},1.25\times {10}^{22}\}$, circles $\{0.0205$, variable, $7.58\times {10}^{-11},8.5\times {10}^{21}\}$, and diamonds $\{0.03$, variable, $7.58\times {10}^{-11},1.05\times {10}^{22}\}$; and (d) squares $\{0.017,18.0$, variable, $\text{variable}\}$, downward triangles $\{0.021,16.0$, variable, $\text{variable}\}$, circles $\{0.0205,18.0$, variable, $\text{variable}\}$, and diamonds $\{0.021,18.0$, variable, $\text{variable}\}$.

Figure 7

Scaling behavior of $r_{1/2}$ with $\sigma$ at various flux rates. For each symbol type the parameters are given as follows $\{\gamma$ (unitless), $D\left({\mathrm{m}}^2{\mathrm{s}}^{-1}\right),F\left({\mathrm{m}}^{-3}{\mathrm{s}}^{-1}\right),F$ (unitless)$\}$: circles $\{18.0,7.58\times {10}^{-11},8.0\times {10}^{21},-0.42\}$, squares $\{18.0,7.58\times {10}^{-11},1.0\times {10}^{22},-0.47\}$, upward triangles $\{18.0,7.58\times {10}^{-11},1.0\times {10}^{22},-0.47\}$ and $\{18.0,7.58\times {10}^{-11},1.2\times {10}^{22},-0.49\}$, and downward triangles $\{18.0,7.58\times {10}^{-11},1.4\times {10}^{22},-0.50\}$.