Diffusive transport of small molecules through a polymer matrix can be described via a statisti-cally random process known as random walk. The driving force behind diffusion is the presence of a gradient concerning the concentration - or more general, the chemical potential - of the diffusing species in the matrix as the system seeks to achieve a homogeneous distribution.280–282 Mathematically, the diffusion of small particles in a surrounding substance can be described via Fick’s law, according to which the net flux J~ of diffusion is proportional to the concentration gradient∇c:280,283
J~=−D· ∇c (6.1)
The proportionality constantD is the so called diffusion coefficient. In general, this coefficient can be also a function of concentrationcand spatial coordinates, if interactions between matrix and diffusing particle are significant or the system is not spatially homogenous.280 In order to describe dynamic processes where diffusion can vary with time, one additionally has to take into account mass conservation via the continuity equation:
∂c
∂t =−∇J~ (6.2)
Inserting this in eq. (6.1) one arrives at Fick’s second law:
∂c
∂t =∇(D· ∇c) (6.3)
In the case of an isotropic system and when D is not concentration dependent, this simplifies to:
∂c
∂t =D·∆c (6.4)
The simplification of assuming an isotropic system in case of a polymeric matrix is justified as long as there is no preferential orientation of the chains that would introduce anisotropy into the mass transport within the system.280Consequently, this simplification can be applied mainly to amorphous polymers. Concentration dependence of the diffusion coefficientDon the other hand is often relevant for polymers when the interaction between the chains and the diffusing species becomes significant,280281 which is strongly dependent on the kind of polymer and the diffusing particle. In the case of fullerenes, these interactions with typical polymers used in organic so-lar cells, like P3HT, PCDTBT or PTB7, become relevant for concentrations so-larger than about 10−20wt%.26,31,57 This in turn means that for small concentrations in the range of only a few wt%, the diffusion coefficient can be regarded as constant. In this case, also the assumption of Fickian diffusion is usually justified, while at larger concentrations deviations may occur.26,280
Wang et al.26 for example found that PCBM diffusion into a PTB7 polymer matrix features two diffusion modes at higher PCBM concentrations (> 40 wt%): A normal Fickian diffusion front of PCBM is followed by a second concentration front of PTB7 saturated with PCBM that evolves linearly with time, i.e. that shows Case-II-like characteristics (=linear dependence of the mass uptake on time284).
Beside concentration, diffusion in polymers depends on several factors such as molecular weight, segmental mobility, side chain structure and functional groups (that might interact with the diffusing particle under certain conditions), as well as π-π-interactions as these influence the aggregation tendency of a polymer.281,285–287 All these factors impact on the temperature de-pendent behaviour of the polymer and its ability to deform and relax under externally applied stress.285,288 A crucial quantity in this respect is the so called glass transition temperature Tg. It is defined as the temperature, where ”the transition from a liquid equilibrium state to a non-ergodic one, i.e. only partially equilibrated state takes place”.288 It is well known that polymers can undergo a transition from a glassy to a rubbery state and finally to a viscous, fluid-like state when temperature is increased.288,289The presence of a rubbery behaviour is determined by the entanglement of polymer chains and therefore structure, molecular weight and chain rigidity. It should be noted, though, that whether a polymer behaves like a glass or a rubber in a relaxation experiment depends on the measurement conditions (measurement time and applied frequency at a certain temperature), meaning that the transition from a glassy to a rubbery state - usually referred to as the ”α-process”1 - is a purely kinetic phenomenon and not an actual structural transition like a melting process. This is directly obvious from the observation that the time range of the glass-rubber transition shifts with temperature.288
The glass transition temperature Tg as such is associated with the occurrence of characteris-tic steps in expansion coefficient dρdT−1 and heat capacity dHdT, respectively. Therefore, common methods to determine Tg are volumetric (DMA) or calorimetric measurements (DSC).285,288 In practice, the characteristic steps in these measurements appear at a temperature where the relaxation time for theα-process is in the order of minutes, i.e. τα(Tg) u 102s.288 This corre-sponds to a heating/cooling rate of 10−2 Ks. For higher rates, the step shifts to shorter relaxation rates and hence slightly higher temperatures. This rate dependence and the fact that the steps are naturally broadened in a real experiment - with typical widths in the order of 10 K - im-plies that Tg should be rather viewed as a temperature range with a tolerance of a few degrees rather than a sharp, well-defined transition temperature. Consequently, for a direct comparison between different materials a fixed heating/cooling rate should be chosen.288
From a microscopic point of view, the observation of steps in dρdT−1 and dHdT is associated with an increased mobility and intensified motion of the chain segments above Tg. This in turn is accompanied by the increase of the so called ”free volume”Vf in the sample, as a rising number of conformational states become populated in which the chains are not or no longer densely packed. Vf is referred to as the volume that is not occupied by the hard cores of the monomer units.288 This concept is usually used to explain the rapid increase in diffusivity above Tg. In case of a mixture like polymer-fullerene blends, the associatedTg may be estimated from the empirical Fox-Flory equation:285,290
1 This process is related to segmental chain relaxation after externally applied stress.288
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1
Tgblend = w1 Tg1 + w2
Tg2 (6.5)
wherew1 and w2 are the weight fractions and Tg1 and Tg2 the glass transition temperatures of the respective component of the blend system. This simple relation does not take into account a possible presence of crystalline domains, but works for anticipating general trends especially in amorphous systems.285,288
It should be emphasized, that Tg as defined above is a quantity that is related to the mobility of the polymer backbone. The side chains are still mobile belowTg and only stiffen at temper-atures well belowTg.285,291 Xie et al. for example have shown that P3HT (rr: Tg ≈ 22 ℃, rra:
Tg ≈ 6 ℃) features a second transition temperature at around−100 ℃ that can be associated with the relaxation of alkyl side chains.291
Above Tg, the viscosity η of a polymer is usually described via the well-established empirical equation known as the Vogel-Fulcher-law:292–294
η(T) =B·exp TA T −TV
(6.6)
where B is a system dependent prefactor, TA denotes the activation temperature and TV is referred to as Vogel temperature. This equation features a singularity atT = TV. Consequently, according to the Stokes-Einstein relation
D = kBT
6πηRv (6.7)
one would expect the diffusion coefficient D of a small spherical particle with hydrodynamic radiusRv to rapidly decrease when approachingTV. TV is typically located 30−70 ℃ below Tg, yet whetherη really diverges seems difficult to be checked by experiment as measurements are only possible down to about 50 K aboveTV, because then η is already very large.288
Concerning diffusion in the vicinity of Tg and below there are several works reporting that dif-fusion of a particle and viscosity of the polymer matrix seem to decouple in a way295–298 and Vogel-Fulcher-Tammann (VFT)-like behaviour of η(T) or equivalently a diffusion coefficient D obeying Stokes-Einstein relation is not necessarily observed forT < Tg.296,299–304 In a rather simple picture, one could think of mass transport rather taking place via thermally activated hopping281,300 like one would expect for interstitial diffusion of atoms or dopants in a solid,305 which is not an inconceivable picture when thinking of a rather stiff and brittle polymer matrix.
Yet, one has to keep in mind that side chains are still mobile leading to continuous redistribu-tion of ”free-volume” in the material.306 The actual underlying mechanism and how to properly describe diffusion below Tg is still subject to research.26,307–310 In order to contribute to the discourse on this issue, this is also addressed in my third publication (see chapters 8.2.2 and 10).
Diffusion in general also plays an important role for device applications in terms of morphological stability of blend mixtures or interfaces between different materials in layered structures.23,25,30,31 311,312Morphology in turn is crucial in terms of device efficiency, for example when considering percolation paths for charge carrier extraction or injection and dissociation or recombination efficiencies (cf. chapters 2 and 3). Consequently, this topic is of special interest to the
or-ganic electronics community and there are several studies dealing with approaches to either suppress diffusion and stabilize morphology23,35,36,274,312–315 or to understand the underlying mechanisms governing miscibility and diffusion in organic electronic devices.25,26,30,57,263,316,317
Special focus in this respect is laid on binary and even ternary blend systems26,31,318 for or-ganic photovoltaics as well as doping of transport layers and at interfaces.319–324 An effective way to reduce diffusion and increase morphological stability is to perform crosslinking of the matrix material or even between two components of a system.23,35,36,274,312–315 This aspect is shortly addressed in chapter 7 and in more detail in the review in appendix A. Some important aspects about diffusion in organic electronic devices resulting from recent research are sum-marized shortly in the following. Several groups found that diffusion mainly takes place in amorphous regions.25,26,30,31,319,325 Crystalline phases are not affected when they have been al-ready present before the mixing,30,326,327but only when they are formed afterwards, for example when sequential deposition is applied.327Furthermore, the degree of intermixing depends on the size and shape of the diffusing molecule, which also provides a means to design molecules for special purposes.319–321,323 For example when using a molecule as a dopant for a transport layer it should be bulky enough to prevent diffusion into the active layer, which in turn may dete-riorate device efficiency. Concerning the relevance of Tg for morphological stability, McEwan et al.328 verified that materials with higherTg tend to diffuse into adjacent layers that feature a lower Tg. So for an overall morphological stability one should only use materials together in a device that all feature a high Tg. Finally, Hartmeier et al.318 emphasize that in complex systems like binary or even ternary blends, in which the components may also tend to crystallize or aggregate, the mixing behaviour is of course not only governed by diffusion kinetics but also thermodynamic aspects. These have to be taken into account when trying to understand the formation of microstructures in these devices.
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