Charge Diffusion in Semiconducting Polymers: Analytical Relation

Letter pubs.acs.org/JPCL

Charge Diffusion in Semiconducting Polymers: Analytical Relation between Polymer Rigidity and Time Scales for Intrachain and Interchain Hopping Paola Carbone*,† and Alessandro Troisi*,‡ †

School of Chemical Engineering and Analytical Science, University of Manchester, Manchester, United Kingdom Department of Chemistry and Centre for Scientific Computing, University of Warwick, Coventry, United Kingdom

‡

ABSTRACT: We study the charge diffusion of semiconducting polymer bulk using simplified coarse grained models to investigate the relation between charge diffusion coefficient and the characteristics time of intrachain and interchain hopping, τ1 and τ2. We consider the process of charge diffusion in several standard models of polymer chains (rigid chain, Gaussian chain, worm-like chain), and we achieve an analytical expression for the diffusion coefficient in terms of the characteristic times and the geometric parameters defining the chain models. The diffusion depends only on the intrachain hopping for the rigid chain and on the geometric average of intrachain and interchain hopping times for the Gaussian chain (the least rigid model), with an analytical interpolation available between two limits. The model highlights the importance of large persistence lengths for improved transport properties. In all cases, it is incorrect to consider the slower interchain hopping as the ratedetermining step for the charge transport. SECTION: Energy Conversion and Storage; Energy and Charge Transport

A

morphous semiconducting polymers1 are different from other disordered inorganic semiconductor (among other things) because they possess an electronic microstructure that favors the charge transport along a chain and makes less favorable the charge transport across different chains. Many successful phenomenological models ignore this property and assume that the charge is transported by independent charge hopping events in all directions.2−4 On the other hand, intuition and more advanced models5,6 suggest that the transport really takes place with two characteristic time scales, that of hopping along a polymer chain and that of hopping between chains. In this note we want to explore how the two time scales contribute jointly to the transport in the material, and we aim to develop very elementary models that can produce analytical results. These results can be useful to rationalize the relation between polymer rigidity and charge transport, which is not much considered among the synthetic design rules that currently guide the development of new materials.7,8 We will see that some results conform to basic intuition, and some others are more surprising and in contrast with commonly encountered statements in literature. This work aims to contribute to the development of a simple set of designs rule for high-performance polymer semiconductors. The main assumption we make in this paper is that the charge transport along a polymer chain can be described as a series of charge hopping events between nearest neighboring “sites” along the chain occurring with a characteristic time τ1. A number of clarifications are important at this point. This is a coarse grained (CG) model where the CG sites do not represent a single monomer but a portion of the polymer chain of length L. The hopping rate 1/τ1 is therefore not the charge © 2014 American Chemical Society

hopping rate between two specific states but an ef fective rate for the charge hopping from any state located in the CG site to any other state located in an adjacent site. It is very well-known that the transport along a one-dimensional disordered chain is not diffusive but dispersive, i.e., many different hopping rates and time scales make it impossible to define a diffusion constant.3 However, if the model is sufficiently coarse, the variability of hopping rates between neighbors can be neglected and the diffusion along the polymer chain is determined by the pair of variables L and τ1 (L can be considered arbitrary, and τ1 adjusted to reproduce the average diffusion coefficient along the chain). We will consider only diffusion and not mobility to avoid the complications introduced by an external field, discussed in ref 9 (where the hopping rate between monomers was assumed to be constant at zero field), and we consider isolated (or noninteracting) charge carriers. We consider the situation where in a characteristic time τ2 ≫ τ1 the charge hops from one chain to another (Figure 1). Since the interchain hopping is so infrequent, the displacement of a charge in a given time will depend very little on the interchain hopping distance. For this reason we can afford not to define a new length scale for the interchain hopping and assume that such hops do not displace the charge by any distance. Given the assumptions above, we want to find out the relation between the macroscopic charge diffusion, D, and the parameters L, τ1, and τ2. We will first consider diffusion along a Received: June 13, 2014 Accepted: July 17, 2014 Published: July 17, 2014 2637

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average square displacement from an initial position, ⟨R2(Δt)⟩RC, is proportional to time Δt ⟨R2(Δt )⟩RC = 2D1DΔt ≡ σΔ2t

(1)

with the 1D diffusion coefficient given by D1D = L2 /τ1

(2)

and the probability distribution of the charge at time t given by f1 (q , t ) = Figure 1. Cartoon representing two coarse grained chains, with site separated by length L, intrachain hopping time τ1, and interchain hopping time τ2.

⎛ q2 ⎞ 1 ⎜− 2 ⎟ exp (2π )1/2 σΔt ⎝ 2σΔt ⎠

(3)

where it was convenient to express the displacement using the continuous variable q and to assume that the charge was located in position q = 0 at t = 0. Such continuous function is an excellent approximation if the time of interest is larger than a few times τ1. If we consider a Gaussian chain model, where the position of the CG sites follows a random walk in 3D, it would be tempting to say that the charge also performs a 3D random walk. In reality, the charge carrier performs a random walk in 1D along the chain’s q coordinate (described by eq 3), but this is folded into the 3D conformation of the chain. The probability distribution in space is the convolution of the 1D Gaussian probability distribution of eq 3 and the 3D Gaussian probability distribution of the GC model (which is itself a Gaussian). We demonstrate below that the charge dynamics in this system does not have the characters of a 3D random walk. In the Gaussian chain model, the average square distance between 2 sites separated by N sites along the chain (end-to-end distance) is ⟨R2⟩GC = NL2.10 If a charge localized on one site at t = 0 is displaced from its initial position on average by ⟨N(Δt)⟩ sites in a time Δt, the average square displacement in the 3D space is given by

Figure 2. Representation of the three models of coarse grained chains considered in this work (rigid chain, RC, Gaussian chain, GC, and worm-like chain, WLC). The scalar coordinate q represents the displacement along the chain (in monomer units) from an arbitrary starting point. The vector R represents the displacement in 3D from an initial position.

single chain and then the combined diffusion resulting from intra- and interchain transport. The results will be dependent on the conformation of the polymer chain, and we will consider three cases widely studied by the polymer physics community and illustrated in Figure 2: (i) Rod-like chain model (RC), where the polymer extends in only one direction and behaves like a rigid rod. (ii) Gaussian chain model (GC), which assumes that the polymer chain is a random walk (ideal chain) of rigid freely jointed subunits of length L, often referred to as Kuhn length. This is the most “flexible” model that can be constructed with rigid segments of length L.10 (iii) Worm-like chain (WLC), used to describe the behavior of semiflexible polymers where a local rigidity is added to the chain through the definition of the chain persistence length, P. This model is of intermediate flexibility and reduces to the RC model for P →∞ and to the GC model when P = L/2.10 The Limit of No Interchain Hopping. It is useful in itself and for the results that follow to consider the case of a charge hopping along a single chain without the possibility of hopping to other chains. This corresponds to the limit of τ2 = ∞. We will use the symbol R for displacement from an initial position (a vector in 3D), R2 for the square of the displacement (a scalar) and q for the curvilinear displacement along the chain. A basic case is that of a rod-like chain, where the problem is identical to the textbook random walk problem in 1D.11 The

⟨R2(Δt )⟩GC = ⟨N (Δt )⟩L2

(4)

N(Δt)L is a positive quantity, corresponding to the average of the absolute value of the displacement, |q|, measured along the chain. Its value can be evaluated analytically (using ∫ ∞ 0 x exp(−ax2) dx = 1/2a) from the distribution in 3 as +∞

⟨N (Δt )L⟩ =

∫−∞

D1DΔt =

|q|f1 (q , Δt ) dq =

2σΔt 2 = π 2π

2 Δt L π τ1

(5)

The average square displacement along a Gaussian chain is therefore ⟨R2(Δt )⟩GC =

2 2 Δt L π τ1

(6)

The motion along a Gaussian chain is clearly subdiffusive with mean square displacement proportional to Δt1/2, while for the rigid rod the mean square displacement is proportional to Δt (simple diffusive transport). A polymer model of intermediate rigidity between the GC and the RC should be able to interpolate between the two limits. In a worm like chain (WLC) with persistence length P and N sites, the average end-to-end distance is12 ⟨R2⟩WLC = 2PNL − 2P 2[1 − exp( −NL /P)] 2638

(7)

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than the persistence length (Δt < P2τ1/L2), the mean square displacement is proportional to time. In the opposite limit of Δt > P2τ1/L2, the mean square displacement is proportional to Δt1/2. The transition takes place when ⟨R2(Δt)⟩ ∼ P2 at time Δt ∼ (P2τ1/L2). Dif f usion with Inter- and Intrachain Hopping. We now consider the situation where in a characteristic time τ2 ≫ τ1, the charge hops from one chain to another. The hops along a chain are so more frequent that we assume that a hopping event between two chains will not be followed by another hopping event that brings the charge back on the original chain. As long as the charge carriers is in one chain it will perform a 1D random walk along the q coordinate of a given chain, which is folded into the (3D) configuration of the polymer chain. When an interchain hop takes place, the charge will perform another similar random walk but along a different polymer chain. The assumption that the carrier will not jump back into the original chain is equivalent to saying that each random walk within a chain is independent from (i.e., uncorrelated to) the path followed while the charge was in the previous chain. Under these conditions the problem can be casted in the following way. • With characteristic time τ2, the carrier hops from one chain to another. • In the interval Δt between interchain hops, the carrier is displaced from the initial position by a distance R(Δt). We could roughly assume that ⟨R2(Δt)⟩ = ⟨R2(τ2)⟩ and evaluate the diffusion coefficient in 3D as ⟨R2(τ2)⟩/6τ2, but this would introduce a small error (except for the rigid chain). We proceed more rigorously below. We consider a large number of interchain hops M, each taking place in a time t1, t2, ..., tM. The probability density of interchain hops for a given value of t is given by the distribution

We proceed as before to compute the average squared displacement as a function of time, by identifying N(Δt)L with the absolute value of the displacement, |q|, and by computing the average using the probability distribution in eq 3. ⟨R2(Δt )⟩WLC =

+∞

∫−∞

{2P|q| − 2P 2[1 − exp( −|q| /P)]}f1

(q , Δt ) dq +∞

+∞

∫−∞ |x|f1 (q , Δt ) dq − 2P 2 ∫−∞ f1 (q , Δt ) dq +∞ + 2P 2 ∫ exp( −|q| /P)f1 (q , Δt ) dq −∞

= 2P

= 2P

⎛ σ2 ⎞ ⎛ ⎛ σ ⎞⎞ 2σΔt − 2P 2·1 + 2P 2 exp⎜ Δt2 ⎟ ·⎜1 − erf⎜ Δt ⎟⎟ ⎝ 2 P ⎠⎠ 2π ⎝ 2P ⎠ ⎝

⎡ 2 D1DΔt ⎛ D Δt ⎞ = 2P ⎢ − P + P exp⎜ 1D2 ⎟ · ⎢ ⎝ P ⎠ π ⎣ ⎛ ⎛ D Δt ⎞⎞⎤ 1D ⎜1 − erf⎜ ⎟⎟⎟⎥ ⎜ ⎜ ⎟ P ⎝ ⎠⎠⎥⎦ ⎝ ⎡ 2 ⎤ = 2P 2⎢ z − 1 + exp(z 2)· (1 − erf(z))⎥ ⎣ π ⎦ (8)

where we have introduced the auxiliary variable z = (D1DΔt)1/2/P = L/P(Δt/τ1)1/2. To further check that the expression above interpolates correctly between the limits of very flexible and very rigid chains we can consider the two most relevant limits. In the limit z → 0 or long persistence length (RC limit) the expression in square brackets is ∼ z2, i.e., ⟨R2(Δt)⟩z→0 = 2D1DΔt (i.e., the results in eq 1). In the opposite limit z → ∞ of short persistence length (i.e., the GC limit where P = L/2) ⟨R2(Δt)⟩z→∞ = (4/√π)P(D1DΔt)1/2 = = (2/ √π)L(D1DΔt)1/2, i.e., the results of eq 6. A plot of the function in eq 8 helps visualizing the transition between the two regimes (Figure 3). When the system is observed for time short enough that that curvilinear displacement along the chain is smaller

Π(t ) =

1 exp( −t /τ2) τ2

(9)

whose average is ∫ ∞ 0 (t/τ2) exp(−t/τ2) dt = /τ2. The total displacement from initial position and for a given set of t1, t2, ..., tM is R(t1 + t 2 + ... + tM ) = R1(t1) + R 2(t 2) + ... + RM(tM ) (10)

The probability distribution for R is normal, being generated by the sum of normal distributions, and the squared average displacement is ⟨R2(t1 + t 2 + ... + tM)⟩ = ⟨R12(t1)⟩ + ⟨R 2 2(t 2)⟩ + ... + ⟨RM 2(tM)⟩

(11)

because ⟨Ri(ti)Rj(tj)⟩ = 0 if the orientation of different chains is uncorrelated. If we take the limit of large M, we have t1 + t2 + ... + tM = Mτ2 (because τ2 is the average hopping time). If we average over all possible choices of times t1, t2, ..., tM we get ⟨R2(Mτ2)⟩ = M

∫0

∞

⟨R2(t )⟩Π(t ) dt

(12)

where the integral indicates the average squared displacement between two consecutive interchain hops. The diffusion coefficient can be computed from the Einstein relation (in three dimensions) as

Figure 3. Relation between mean squared displacement and scaled time for a worm-like chain, illustrating the transition between diffusive and subdiffusive behavior when ⟨R2(Δt)⟩ ∼ P2. 2639

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⟨R2(Mτ2)⟩ 1 = 6Mτ2 6τ2

∫0

∞

Letter

⎤ ⎡1 ⟨R2(t )⟩⎢ exp( −t /τ2)⎥ dt ⎦ ⎣ τ2 (13)

The integral can be readily solved for different types of chains possessing different ⟨R2(t)⟩ functions. For rod-like chain model, where ⟨R2(t)⟩RC = (2L2τ1)t, we have simply DRC =

1 2L2 6τ2 τ1

∫0

∞

t

1 L2 exp( −t /τ2) dt = τ2 3τ1

(14)

For Gaussian chain model, where ⟨R2(t)⟩GC = (2√π)L2(t/ τ1)1/2, we have DGC = =

1 2 2 1 L τ1 6τ2 π

∫0

∞

t 1/2

1 exp( −t /τ2) dt τ2

1 2 2 1 π 1/2 L2 τ2 = L τ1 2 6τ2 π 6 τ1τ2

(15)

The difference between the two cases is remarkable. In fully rigid chain, the time scale for interchain hopping is completely irrelevant while for the Gaussian chain the relevant time scale is the geometric average between the interchain and the intrachain hopping time scale. As we have assumed that τ1 ≪ τ2, the diffusion coefficient is much larger for rigid chains (or it increases with the increased rigidity of the chain). An interpolation between the two limits can be achieved by substituting eq 8 into eq 13. The result, which in this case does not have a simple analytical form, is illustrated by Figure 4(top) where the relation between diffusion coefficient and polymer persistence length is given for the cases τ2 = 16τ1, 64τ1, 144τ1. The figure illustrates how the characteristic time τ2 is important at low persistence lengths and becomes irrelevant at high persistence lengths. To better clarify the conditions that makes the worm-like model behave as a rigid rod we recall the observation made in Figure 3 that the mean square displacement along a chain is that observed for a rigid rod until a critical time (P2τ1/L2). As τ2 is the average residence time on a chain before hopping to a different chain, the transition between the two diffusion regimes is expected at τ2 ∼ (P2τ1/L2). This suggests that the persistence length P can be scaled by L(τ2/τ1)1/2 to obtain a universal relation between suitable scaled diffusion coefficient and persistence length. Inspection of eqs 8 and 13 confirms that there is a universal relation between the diffusion coefficient scaled by L2/τ1 and the persistence length scaled by L(τ2/τ1)1/2, which is reported in Figure 4 (bottom). The very simple relations obtained rely very strongly on the fact that the average distance traveled on a chain depend only on the parameters L and τ1. The inclusion of an electric field substantially complicates the situation because the distance traveled will depend on the orientation of the chain and the field strength. Moreover, kinks of the chain in the direction of the field may act as traps for the carrier,9 and we do not expect that the scaling relations illustrated above remain valid in the presence of electric fields. The main advantage of the models presented here is that they provide simple analytical expressions containing very intuitive parameters. Such models suggest important clarifications on the interplay between intrachain and interchain transport. For example, it is sometime assumed that, since the intrachain hopping rate is much smaller than the interchain, the

Figure 4. (Top) Dependence of the diffusion coefficient on the persistence length for different values of τ2/τ1. (Bottom) Universal relation between scaled diffusion coefficient and persistence length in the work-like chain.

overall transport is determined by the interchain hopping, which can be considered the rate-determining step. The analytical results show on the contrary that the interchain hopping time is at most as important as the intrachain transport, and it becomes less relevant only for relatively rigid chains (high persistence length). Such an observation implies that, when quantum chemical models are constructed,5,13,14 the intra- and interchain hopping rates should be given the same importance. Furthermore, one of the least understood aspects of polymeric semiconductor is the relation between their morphology and transport properties. Processing15,16 and synthetic modifications17 can significantly alter the polymer persistence length, especially when polymers aggregate or form partially crystalline domains.18 The model suggests a substantial enhancement of the transport characteristics for more rigid chains. As we mentioned in the introduction, the parameters L and τ1 are meant to describe in some effective way the transport along the chain and should not be confused with an average hopping length or time of actual charge transfer events. Individual hopping events are appropriately described in an energetically disordered potential energy landscape,19 with individual hopping rates influenced by different charge localizations,20 in turn determined by the disorder in the 2640

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torsional degrees of freedoms.21 The effect of disorder within this model is therefore, in principle, included in the characteristic times τ1 and τ2, i.e., larger disorder would result in reduced hopping times. In reality, however, the effective hopping time between two adjacent polymer portions (sites) will always be given by some distribution of times rather than a single characteristic time. The model breaks down if this distribution is very broad, most typically if there is a finite chance of very long hopping time between two sites, for example, corresponding to poor electronic coupling between the sites. The model could be generalized to incorporate this effect, for example, by allowing the chain of being of finite lengths (i.e., the charge cannot hop on a given chain beyond a certain point). Any additional effect preventing diffusion along the chain, including the presence of an external field, will inevitably increase the importance of time τ2 for any model of chain, including the rigid chain. In this work, however, we preferred to focus on models that are analytically tractable, which can be the starting point for numerical studies of more detailed models.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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