Temperature-Dependence of Persistence Length Affects

Article Cite This: Chem. Mater. 2018, 30, 748−761

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Temperature-Dependence of Persistence Length Affects Phenomenological Descriptions of Aligning Interactions in Nematic Semiconducting Polymers Jaime Martin,*,†,‡ Emily C. Davidson,¶ Cristina Greco,§ Wenmin Xu,∥ James H. Bannock,∥ Amaia Agirre,† John de Mello,∥ Rachel A. Segalman,*,¶,⊥,# Natalie Stingelin,*,⊗ and Kostas Ch. Daoulas*,§ †

POLYMAT, University of the Basque Country UPV/EHU, Avenida de Tolosa 72, 20018 Donostia-San Sebastián, Spain Centre for Plastic Electronics and Department of Materials, Imperial College London, Exhibition Road, London SW7 2AZ, United Kingdom ¶ Department of Chemical Engineering, University of California Santa Barbara, Santa Barbara, California 93106, United States § Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany ∥ Centre for Plastic Electronics and Department of Chemistry, Imperial College London, Exhibition Road, London SW7 2AZ, United Kingdom ⊥ Materials Department, University of California Santa Barbara, Santa Barbara, California 93106, United States # Mitsubishi Centre for Advanced Materials, University of California Santa Barbara, Santa Barbara, California 93106, United States ⊗ School of Materials Science and Engineering and School of Chemical & Biomolecular Engineering, Georgia Institute of Technology, 311 Ferst Drive, Atlanta, Georgia 30332, United States ‡

S Supporting Information *

ABSTRACT: Electronic and optical properties of conjugated polymers are strongly affected by their solid-state microstructure. In nematic polymers, mesoscopic order and structure can be theoretically understood using Maier−Saupe (MS) models, motivating us to apply them to conjugated macromolecular systems and consider the problem of their material-specific parametrization. MS models represent polymers by worm-like chains (WLC) and can describe collective polymer alignment through anisotropic MS interactions. Their strength is controlled by a phenomenological temperaturedependent parameter, υ(T). We undertake the challenging task of estimating material-specific υ(T), combining experiments and Self-Consistent Field theory (SCFT). Considering three different materials and a spectrum of molecular weights, we cover the cases of rod-like, semiflexible, and flexible conjugated polymers. The temperature of the isotropic−nematic transition, TIN, is identified via polarized optical microscopy and spectroscopy. The polymers are mapped on WLC with temperature-dependent persistence length. Fixed persistence lengths are also considered, reproducing situations addressed in earlier studies. We estimate υ(T) by matching TIN in experiments and SCFT treatment of the MS model. An important conclusion is that accounting explicitly for changes of persistence length with temperature has significant qualitative effects on υ(T). We moreover correlate our findings with earlier discussions on the thermodynamic nature of phenomenological MS interactions.

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INTRODUCTION

molecular conductivity leads to enhanced charge mobility on macroscopic scales. Specially designed processing protocols are among basic methods used to manipulate chain alignment in conjugated polymers. One approach is to achieve favorable order by locking14 the morphology in a nonequilibrium state. Strain alignment,15 mechanical rubbing,9,16 epitaxial crystallization,7

Strategies promoting long-range molecular alignment in semiconducting (conjugated) polymers attract significant attention due to improved electronic and optical properties of such morphologies.1−11 For example, elongated chain conformations without “twists” (acting as defects1,3 interrupting conjugation) allow for fast delocalization of charges along molecular backbones. In combination with some events3,12,13 of π-stacking between neighboring backbones, which are essential for efficient intermolecular charge transport, high intra© 2018 American Chemical Society

Received: October 5, 2017 Revised: January 16, 2018 Published: January 17, 2018 748

DOI: 10.1021/acs.chemmater.7b04194 Chem. Mater. 2018, 30, 748−761

Article

Chemistry of Materials

energies and the parameters are generally state-dependent, e.g., the MS parameter depends on temperature, T. Once the parameters are specified, the properties can be explored through various computational techniques: from numerical Self Consistent Field (SCF) theory47,50−58 to particle-based simulations.34,37,38,59,60 So far, the schemes used to parametrize MS models are topdown. Namely, the parameters are adjusted so that the MS model reproduces certain observables known from experiments or atomistic simulations. For instance, the parameters of the WLC can be chosen so that it recovers the persistence length in Θ-solvent conditions as well as the contour length of the actual polymer. The former can be obtained either experimentally, e.g., scattering,61 or from simulations.36 Quantifying υ is the most challenging task, and for this purpose two strategies have been developed. One possibility is to choose υ such that the temperature of the isotropic−nematic (IN) transition, TIN, in the MS model and experiments match. Chain stiffness correlates orientations of segments in the same molecule. Because these correlations are synergistic to alignment forces between individual segments, increasing chain length shifts TIN to higher temperatures.25,62 For long chains (comparing to the persistence length), this effect approaches saturation.36,62−64 The temperature dependence, υ(T), of the MS parameter can be estimated from the spectrum of values of υ required to reproduce TIN for different chain lengths. This method was first applied32 to poly(2,5-di(2′-ethylhexyloxy)-1,4-phenylenevinylene) (DEHPPV), treating these stiff polymers as rigid rods. In the rigidrod limit, mean-field treatment leads to υ(TIN)/kTIN, which is inversely proportional to chain length (k is the Boltzmann constant). Based on this result, υ(T)/kT was estimated for DEH-PPV and was found32 to be a decreasing function of temperature. Two flexible polyalkylthiophenes, poly(3-dodecylthiophene) (P3DDT) and poly(3-(2′-ethyl)hexylthiophene) (P3EHT), have been also approximated25 by the rigid-rod model, delivering MS parameters with similar temperature dependence. Recently, another strategy has been proposed.36 The method combines all-atom simulations and SCF calculations to estimate the strength of aligning interactions, even when relevant IN transitions are experimentally inaccessible. In the all-atom simulations melts of conjugated polymers are considered at elevated temperatures and weak nematic order is induced by external tension. The same setup is considered in SCF calculations with a MS model based on a semiflexible WLC, so that υ is established by matching the magnitude of the order parameter in simulations and SCF. The approach was applied36 to poly(3-hexylthiophene)(P3HT) melts and the temperature dependence of the MS parameter, followed qualitatively the trends reported in the studies of DEH-PPV, P3EHT, and P3DDT summarized above. The atomistic simulations of P3HT melts were conducted at temperatures equal of higher than 600 K. The analysis of these simulations justified the implementation of a single value of persistence length for all considered temperatures. Because intramolecular correlations in segmental orientation influence the IN transition, assumptions made on molecular architecture in MS models will affect top-down estimations of the MS parameter. Approximating flexible molecules by rigid rods25 cannot capture weakening of orientational correlations along the contour of a long chain, leading to saturation of TIN at large molecular weights (MW). A MS parameter estimated by

and sandwich casting which combines capillary action with textured substrates5,6 are typical examples. An alternative strategy14,17 is to template well ordered solid-state morphologies by processing the material in a liquid phase where chain alignment is thermodynamically favored. The approach follows the general ideas18−20 of molecular self-assembly, broadly used in other areas of nanotechnology. Creating the desired morphologies through the thermodynamic route has advantages such as reproducibility and scalability. Liquid crystalline (LC) mesophases of semiconducting polymers are an example of a thermodynamically stable liquid state with high degree of molecular order. Several studies have demonstrated that such mesophases can indeed serve8,10,21−24 as a template for solid-state microstructures with favorable characteristics, especially for molecular architectures designed to improve4,10,14,17,23 π-stacking. At the same time, not all conjugated polymers exhibit LC mesophases. One of the reasons is the proximity of the melting temperature to the thermal degradation.25,26 Therefore, there is significant interest in designing semiconducting polymers with accessible LC mesophases.14 To obtain morphologies with desired structure, liquid crystallinity can be used synergistically with other ordering processes. The interplay between LC ordering and microphase separation in block copolymers incorporating conjugated blocks is an example.27−31 In conjugated polymers, mesoscopic properties of morphologies with uniaxial (nematic) LC order can be theoretically understood using Maier−Saupe (MS) models.32−38 Approaches based on MS models have been successful in describing nematic mesophases for a broad range of molecular chemistries, e.g., from rod-like derivatives of poly(alkoxyphenylenevinylene) 32 to flexible polyalkylthiophenes.25,36 MS descriptions belong to the class of minimal39 (or generic40,41) models of polymer physics. Minimal models represent molecular architecture and interactions in simplified manner, retaining only some key mesoscopic features. They are deeply rooted39,40 in universalities characterizing the longwavelength behavior of various classes of polymer systems.42 Following the ideas of universality, MS models describe the architecture of conjugated polymers through simple worm-like chain (WLC) representations. The effect of microscopic factors favoring nematic alignment in the actual material, is represented by a phenomenological anisotropic nonbonded potential. The latter is pairwise and short-ranged, with an anisotropic part proportional to −υP2(cos(θ)) . Here θ is the angle between two interacting WLC segments and P2 stands for the secondorder Legendre polynomial. The parameter υ > 0 sets the strength of MS interactions. MS models can be incompressible or compressible, in which case the anisotropic interactions are augmented by simple, isotropic repulsive potentials.43,44 For multicomponent systems, additional isotropic interactions are included44−46 to account for incompatibility of different monomers. Although MS models have been initially applied to thermotropic nematics, compressible variants are also applicable to lyotropic systems.47 In fact, the standard Onsager model48 for lyotropic nematics can be approximately47,49 reduced to a compressible MS description. Though the functional-forms of the bonded and nonbonded interactions in the MS model are generic, the parameters controlling the stiffness of the WLC and the strength of interactions are chemistry-specific. Establishing these parameters is a prerequisite for material-specific predictions. Because the MS model is mesoscopic, its “potentials” are in fact free 749


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P3EHT polymers have been synthesized via Grignard metathesis polymerization and details can be found elsewhere.69,70 The numberaveraged molecular weights, Mn(α), degree of polymerization, Ñ α, and dispersity (ratio of weight-average and number-average molecular weights), Đα, of PFO and P3EHT are summarized in Tables 1 and 2

matching TIN at different MW will implicitly incorporate these chain-length effects, making υ(T) also MW-dependent. This effective MW-dependence confounds the use of the MS parameter as a quantifier of segmental interactions. In contrast, SCF theory calculations36,37,63,64 based on WLC with finite flexibility describe explicitly polymer conformations and capture explicitly effects of molecular weight related to intramolecular correlations in segmental orientation. Therefore, top-down estimations of MS parameters in SCF are free from effective MW-dependencies originating from intramolecular correlations. In SCF calculations, it is straightforward to study cases where the flexibility of the molecule is affected by temperature. Using WLC models with constant persistence length is justified for rigid polymers or flexible polymers exhibiting an IN transition at temperatures high enough to maximize conformational freedom. However, for conjugated polymers where the IN transition occurs at lower temperatures this is not a valid assumption. For instance, in P3EHT systems with moderate chain lengths IN transitions occur25 below 400 K. In this range of TIN, the flexibility of poly(alkylthiophenes) changes noticeably with temperature, as demonstrated by recent scattering experiments.61 Hence, for certain materials, the MS parameter estimated with models where persistent length is constant, will implicitly include effects on IN transition related to changes of chain flexibility with temperature. Such effects will be excluded from MS parameters estimated with models taking explicitly into account the influence of temperature on polymer stiffness. In this work, we address the important, basic question how estimations of MS parameters can be qualitatively affected by using models with temperature-dependent persistence length. For this purpose, we begin with considering two conjugated polymers with very different flexibilities, forming nematic mesophases:25,65 the semiflexible poly(9,9-di-n-octylfluorenyl2,7-diyl) (PFO) and the flexible P3EHT. For both materials, several molecular weights are considered, for which we experimentally locate TIN via polarized optical microscopy and spectroscopy. Subsequently, we introduce a MS model where PFO and P3EHT molecules with molecular weights studied in the experiments are represented by a discrete WLC. The WLC is parametrized using experimental data,61,65 to reproduce temperature-dependent persistent lengths of PFO and P3EHT. For different chain lengths, the thermodynamics of the MS model is described within SCF theory based on partial enumeration37,66−68 and we identify the values of the MS parameter for which TIN in the MS model and experiments match. Similarly to previous SCF studies with continuum WLC,36,63,64 our calculations describe explicitly polymer conformations. To extend the spectrum of investigated molecular flexibilities, we also benefit from older data32 obtained for short rod-like chains of DEH-PPV. Our results demonstrate that describing explicitly the temperature dependence of the persistence length has striking qualitative effects on the temperature dependence of the extracted MS parameter. This result has important practical implications since the temperature dependence of the MS parameter can provide molecular-design guidelines for achieving macroscopic polymer alignment.14

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Table 1. First Four Columns: Nomenclature and Characterization of PFO Samples in Experiments; Last Column: Number of Segments in the Discrete Worm-like Chain Used to Represent These Samples in SCF Calculations sample

Mn(PFO) (kg/mol)

Ñ PFO

ĐPFO

NPFO

PFO-21 PFO-26 PFO-28 PFO-33

8 10 10.8 12.8

21 26 28 33

1.1 1.4 1.1 1.4

10 − 14 16

Table 2. First Four Columns: Nomenclature and Characterization of P3EHT Samples in Experiments; Last Column: Number of Segments in the Discrete Worm-like Chain Used to Represent These Samples in SCF Calculations sample

Mn(P3EHT) (kg/mol)

Ñ P3EHT

ĐP3EHT

NP3EHT

P3EHT-31 P3EHT-37 P3EHT-54 P3EHT-65 P3EHT-181

5.6 6.7 9.8 11.7 33

31 37 54 65 181

1.2 1.2 1.2 1.2 1.4

15 18 26 32 −

respectively. The subscript α denotes the chemical species of the polymer (PFO or P3EHT). Details concerning the characterization of chain length are provided in the Supporting Information. Polarized Optical Microscopy and Spectroscopy. The TIN of PFO samples and P3EHT-181 were experimentally determined using a method based on polarized optical microscopy and spectroscopy (POM-S). For the measurements, samples were positioned in a microscope hot stage (Linkam Scientific Instruments Ltd.) in the light path between the polarizer and the analyzer of an Olympus BX51 microscope configured in crossed polarizers mode. Thus, in the absence of any sample or when an isotropic material is placed no light passes through the analyzer and reaches neither the spectrometer (Ocean Optics USB2000+) nor the camera (Canon 5D Mark II), which are positioned at the end of the light path, just after the analyzer. However, optically anisotropic materials, such as polymer nematics, change the state of polarization of light, causing the light to pass through the analyzer and to reach both the spectrometer and the camera. Within the hot stage, the polymer pieces were sandwiched between a glass slide and glass coverslip. The coverslip was very gently placed over the molten polymer samples in order not to exert strong pressures that reduce excessively the sample thickness. It is well-known that interfaces can promote the nematic alignment of molecules. Hence our aim was to determine the TIN of bulk-like polymers so that it can be employed as a reference in SCF calculations where interfaces are absent. Therefore, the typical thickness of our samples was 30−50 μm. To determine TIN, the samples were heated at a constant rate while images and spectra (in visible wavelength region) were simultaneously collected. A heating rate of 0.05 °C/min was applied in order to limit the impact of kinetic effects on the IN transition, so that the extracted TIN characterizes as well as possible the true thermodynamic transition (which is the one addressed in SCF calculations). With this requirement in mind, our experiments were programmed to acquire a spectrum every 1 °C and 20 min. The transmitted intensity at each temperature was then integrated between the suitable wavelength

MATERIALS AND EXPERIMENTAL METHODS

Polymers and Their Characterization. Experiments were performed on poly(9,9-di-n-octylfluorenyl-2,7-diyl) (PFO) and poly(3-(2′-ethyl)hexylthiophene) (P3EHT) polymers. The PFO and 750


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Chemistry of Materials Nα − 1

range (480 to 700 nm) to obtain the value of the total intensity transmitted through the materials. Because photons reach the spectrometer solely because the nematic phase is birefringent, the normalized value of the total intensity transmitted is related to the relative amount of nematic phase present at each temperature. Thus, a conversion index for the isotropic−nematic transition, C.I., can be defined as the normalized value of the integral of the transmitted light intensity. When the polymers are in the nematic phase, at temperatures well below TIN, the light passing through the material is maximum and C.I. = 1. Conversely, when the materials are in the isotropic phase, C.I. = 0. Hence, monitoring C.I. as a function of temperature, enables us to follow the melting of the nematic phase into the isotropic phase. By analogy with how thermal transitions are typically defined in a differential scanning calorimetry (DSC) experiment, in this work we define the TIN as the temperature at which the C.I. changes more rapidly, i.e., the temperature at which the relative amount of nematic phase disappears more rapidly (|d(C.I.)/dT| has a maximum). For further details about the measurement methods employed for the assessment of TIN, see the Supporting Information. We point out that in this study, solely the transition from the nematic state to the isotropic state with increasing temperature is considered. The reason is 2-fold. On the one hand, in order to assess the IN transition during cooling, the polymers must be taken to very high temperatures in order to have a purely isotropic melt (especially the PFO), which may induce thermo-oxidative degradation. On the other hand, though the nematic-to-isotropic transition (i.e., the transition during heating) can be generally assumed to be a thermodynamically controlled process, at least when slow heating rates are applied, the isotropic-to-nematic transition (i.e., the transition during cooling) is expected to be affected by kinetic factors. Indeed, strong hysteretic behavior is found for our PFO system, suggesting that kinetic factors hamper molecular ordering. Note that this is a general behavior of phase transitions in polymers, where the formation of a certain ordered phase upon cooling frequently occurs at temperatures far below the destruction of the same phase upon heating (for example, crystallization and melting). For systems with small mesogens simulations studying formation of nematic phase from an isotropic phase have been performed.71 Two pathways have been identified: nucleation and growth, as well as spinodal-like decomposition in analogy to systems with conserved order parameters.72 In the spinodal-like decomposition mechanism, the system develops rapidly a large number of small nematic domains with different orientations, adjacent to each other in a labyrinth structure.71 However, developing long-range nematic order extends over longer times as it requires alignment of these small domains. It is expected34 that for polymers, such domain alignment will be even more protracted due to slow chain dynamics. Therefore, although we employ throughout the paper the term isotropic−nematic transition, our experiments consider only the heating branch, i.e., we monitor the transition only from nematic to the isotropic state. These nematic samples can still contain rather small domains with different orientations. Hence, using transmittance measurements under crosspolarizers is an advantage (e.g., classical textures expected by microscopy are harder to detect). The TIN of P3EHT polymers other than P3EHT-181 is deduced by POM, defined as the temperature at which the birefringent textures disappear. These data have been reported elsewhere.25

Hb(α) = −εα

∑

u i(s) ·u i(s + 1)

(1)

s=1

where the subscript α denotes the chemical species of the polymer (PFO and P3EHT) and ui(s) is a unit vector along the s-th segment of the i-th WLC. For both PFO and P3EHT, we assume that a single WLC segment represents a pair of repeat units, i.e., two fluorenes or two (2′-ethyl)hexylthiophenes. The details of mapping atomistic monomers on a single WLC segment are clarified in Figure 1a,b for PFO and P3EHT

Figure 1. (a) Scheme used to map PFO (top) and P3EHT (bottom) chains on the discrete WLC model. In both cases, one WLC segment represents two actual repeat units and has length bα. For clarity, the sections of the polymers and the WLC are shown in an all-trans (fully stretched) conformation. (b) Sketch of a discrete WLC in a random conformation, summarizing the parameters used to define the chain architecture.

polymers, respectively. The length of the WLC segment, bα, presents the first physically relevant length scale described by the discrete WLC model. On the basis of geometrical parameters for PFO monomers73 and poly(alkylthiophenes),74 we estimate bPFO = 1.67 nm and bP3EHT = 0.79 nm. The persistence length, lp(α)(T), of polymers at different temperatures, T, presents the second physical length scale conserved by the model through an appropriate choice of the stiffness parameter, εα. In this study, in eq 1 we consider a constant (state-independent) stiffness parameter which is the simplest way of obtaining a WLC model with temperaturedependent persistence length (it is reminded that the conformational Boltzmann weight is determined by the ratio Hb(α)/kT). To estimate εα we take into account that the persistence length of an ideal WLC is determined by75 lp(α)(T ) =

bα ⎡ ⎤ 2ε 1 − exp(− kTα ) ⎥ ln⎢ kT ⎢⎣ 1 − ε + (1 + kTε )exp(− 2kTεα ) ⎥⎦ α α

(2)

Substituting the persistence length of the actual polymer known experimentally for some temperature, T0 (α), in Θ-solvent conditions, we solve eq 2 numerically to obtain εα. Figure 1b summarizes the parameters used to define the architecture of a WLC. For PFO, we employ as a reference lp(PFO) = 8.5 nm at T0(PFO) = 40 °C based on data reported by Grell et al.65 For P3EHT, we set lp(P3EHT) = 3.15 nm, which presents the average of persistence lengths obtained61 at room temperature, T0(P3EHT) = 22 °C, from small-angle neutron scattering experiments near Θ-solvent conditions. The values εPFO =

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MODEL AND SELF CONSISTENT FIELD THEORY APPROACH Model. For the coarse-grained representation of PFO and P3EHT polymer molecules, we employ the discrete WLC model. Namely, for a system with nα identical chains the Hamiltonian of bonded interactions for the i-th molecule is given by 751


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Chemistry of Materials 2423.46 × 10−23 J and εP3EHT = 1835.06 × 10−23 J calculated from these reference data, can be substituted into eq 2 to obtain the temperature dependence of the persistence length, presented in the main panel of Figure 2. The predictions of

panel of Figure 2 (solid squares) experimental data for the lp of P3HT (another poly(alkylthiophene)) reported as a function of temperature in an earlier study.61 We observe that the experimental data are reasonably close to the simple WLCbased estimate (parametrized from a single temperature point). The variations of persistence length observed in the WLC model of P3EHT and in the experiments are comparable, in a similar range of temperatures. Currently, there is considerable interest77 in predicting the persistence length of conjugated polymers theoretically. Although such predictions can be obtained through all-atom Monte Carlo or Molecular Dynamics simulations, scanning a broad-range of system parameters is demanding computationally and requires high-quality force fields. The persistence length can be efficiently estimated through single-chain calculations,61,77,78 where conjugated polymers are mapped on variants of the hindered rotation (HR) chain model. Deflection and torsional angles of the HR chain are controlled by angular and dihedral potentials, Uθ(θ) and Uϕ(ϕ). The persistence length can be estimated either analytically,78 using an elegant transfer-matrix method, or from stochastic sampling78 of HR chain conformations. Specifying Uθ(θ) and Uϕ(ϕ) is challenging. Although useful estimates can be obtained from DFT calculations, Uθ(θ) and Uϕ(ϕ) are effective coarse-grained potentials that implicitly account for nonbonded interactions in the multichain environment in the real material. Therefore, it is preferable to derive Uθ(θ) and Uϕ(ϕ) by standard coarse-graining79 of reference all-atom configurations. However, in this case Uθ(θ) and Uϕ(ϕ) are state dependent.79 For external conditions considerably different than the reference state point, these potentials lead to less accurate predictions. It is instructive to obtain a simple estimate of temperature effects on the persistence length of poly(alkylthiophenes) using the HR chain model. For this purpose, we benefit from an effective Uϕ(ϕ) obtained78 on the basis of all-atom simulations of a melt of P3HT 20-mers at 700 K. The polynomial Uϕ(ϕ) = ∑i5= 0ci cosi(ϕ) with c0 = 8.904, c1 = −2.285, c2 = −19.346, c3 = 3.365, c4 = 13.767, and c5 = −2.847 (all ci are given in kJ/mol) reproduces closely the data for the effective Uϕ(ϕ) reported in that study. The bond length of the HR chain is set to l = 0.4 nm. Because an effective Uθ(θ) is not available to us, we fix the deflection angle to θ = 32.6°. On the basis of this simple “force field”, we sample the conformations of the HR chain with standard Monte Carlo80 and obtain at 700 K lp = 3.26 nm. This value matches the lp reported in the all-atom simulations,78 justifying our choice of θ. The inset of Figure 2 presents the lp estimated with this approach in a temperature range relevant to our study. These lp values are larger than the persistence length in experiments and the WLC model (based on top-down parametrization). Bearing in mind the transferability issues, this quantitative difference is not surprising: the data in the inset refer to temperatures that are about two times smaller comparing to the 700 K at which Uϕ(ϕ) was obtained. Moreover, Uϕ(ϕ) was derived from melts, whereas the experiments were performed in marginal solvent. Additional implications might be linked to transferability of the atomistic force fields underlying the Uϕ(ϕ). However, as far as relative changes are concerned, the HR chain model predicts a ∼3.8% variation. This estimate is comparable to the ∼6% changes considered for P3EHT in the main part of our study. Nonbonded interactions are described within a MS model, assuming that the polymer system is incompressible with

Figure 2. Main panel: Dependence of persistence length, lp(α), on temperature for α = PFO (black line) and P3EHT (red line). The open symbols (square and circle) mark the value of lp(α) known from experiments (at given temperature) used to parametrize the WLC model. Solid squares are experimental data61 obtained for the temperature dependence of persistence length for P3HT. Black and red arrows mark for PFO and P3EHT (respectively) the temperature range that is relevant for our study. Inset: temperature dependence of persistence length, lp, for P3HT estimated from a simple HR chain model (details provided in text).

the WLC model for PFO and P3EHT are shown with black and red lines, whereas the open symbols (circle and square, respectively) mark the reference point. Due to the coarsegraining scheme the number of segments in the WLC, Nα, is about half the number of repeat units in the actual PFO or P3EHT molecule (the precise Nα used to represent each sample can be found in Tables 1 and 2). Thus, the WLC model reproduces a third physical length scale: the contour length of the actual polymer, Lα = Nαbα (defined as the maximum end-toend distance of the molecule). The black and red vertical arrows in the main panel of Figure 2 mark for PFO and P3EHT (respectively) the range of temperatures that are relevant for our study. It can be observed that the corresponding persistence lengths for PFO and P3EHT vary from lp(PFO) ≃ 5.56 to 4.68 nm and lp(P3EHT) ≃ 2.53 to 2.37 nm, respectively. This substantial change of persistence length with temperature (∼16% for PFO and ∼6% for P3EHT) affects qualitatively the isotropic−nematic (IN) transition, as will be discussed in the Results section. In general, for some polymers, reproducing accurately with a WLC model the persistence lengths measured in experiments at different temperatures, might require the implementation of a temperature-dependent stiffness parameter, εα(T) . This state dependence is ultimately linked to the free-energy character of the effective potential Hb(α) in the mesoscopic WLC model. For instance, temperature-dependent stiffness parameters have been considered when employing a WLC model to represent complex polymers such as DNA.76 WLC with temperaturedependent stiffness parameters can be straightforwardly handled within our theoretical framework. At the same time, we emphasize that the simple approximation εα(T) ≃ εα adopted in this study leads to variations of persistence length with temperature that are qualitatively comparable with certain experimental data. As an illustration, we reproduce in the main 752


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Chemistry of Materials density of coarse-grained segments ρ0(α) = nαNα/V (where V is the volume of the sample). Because the density is fixed by the incompressibility constraint, only anisotropic interactions between segments are explicitly introduced, through the Hamiltonian: Hnb(α) = −

1 2

isotropic and nematic mesophases within the incompressible MS model, we must introduce a single tensorial mean field, W. This field does not depend on the position in space and expresses the anisotropic MS interactions. Formally, there is an additional scalar field conjugated to the local density which can be seen as a Lagrange multiplier enforcing the incompressibility constraint.83 However, for our homogeneous systems, this field presents a constant which can be omitted. The mean field W can be obtained taking into account eq 3 as

nα , Nα nα , Nα

2υα(T ) δ(rij(s , t ))q i(s): q j(t ) 3ρ0(α) j,t=1

∑ ∑′ i,s=1

(3)

The prime symbol indicates that the second summation excludes self-interactions (i.e., s ≠ t when i = j), whereas rij(s,t) = ri(s) − rj(t) . Here, ri(s) and rj(t) are the coordinates of the center of mass of the s-th and t-th segment along the i-th and j-th chain, respectively. The eq 3 assumes that the range of interactions is smaller than any other physically relevant length scale so that the potential between two segments is proportional to the δ-function. The symmetric traceless tensor, quantifying the orientation of a segment, is defined as q i(s) =

3 I u i (s ) ⊗ u i (s ) − 2 2

nα , Nα

∑

W=−

j,t=1

2υ δ(rij(s , t ))q j(t ) 3ρ0(α)

=−

2υ q 3

(5)

where angular brackets denote an average in the canonical ensemble and the averaged quantity has the meaning of an instantaneous field acting on a segment in a single configuration. The average orientation tensor is given by q = ⟨qj(t)⟩. Within the SCF framework, q is obtained from conformational statistics of single chains in the field W as

(4)

where I is the unit matrix. Due to locality of interactions and incompressibility, the model does not include any effects of local liquid packing present in actual materials. The potential in eq 3 presents as a phenomenological choice with minimum symmetry required to bring-up uniaxial nematic order. It is helpful to see this form as a simple approximation of a complex, unknown, effective potential which can describe accurately the interactions between coarse-grained segments with cylindrical symmetry. The expansion81,82 of this unknown potential into series based on Wigner rotation matrices contains the MS form of eq 3 (assuming locality of interactions). Here, in contrast to compressible MS models, the isotropic term of such an expansion is neglected due to the incompressibility constraint. The explicit dependence of the parameter, υα(T), on temperature highlights the free-energy nature of the coarsegrained MS potential. Self Consistent Field Theory Based on Partial Enumeration of Conformations. In our approach for estimating υα(T), it is important to determine the Helmholtz free energies of the isotropic and the nematic phases at fixed temperature, as a function of υ. Because in this description of the free-energy landscape υ stands for a free parameter, in this section the indication of temperature dependence and the material-specific subscript are dropped off. The isotropic and nematic phases are considered as homogeneous, e.g., there are no local variations in the nematic order. The required freeenergy landscape, follows from an approximate treatment of the statistical mechanics of the MS model within standard SCF theory. The SCF theory formalism for MS-type models of polymer nematics has been extensively presented in the literature.37,50−58,63 Hence, here we provide only a brief discussion of the basic SCF results, relevant for our study. To clarify better the underlying physics, we avoid a formal fieldtheoretical treatment and introduce the SCF framework through simple arguments. SCF theory replaces a system of interacting polymers with an ensemble of independent chains in external fields. These mean fields represent the nonbonded interactions between segments and are functions of certain observables expressed through average properties of single chains. Self-consistency is obtained by linking these average properties to conformational statistics of chains in the fields. Here, to describe the homogeneous

1 q= N

N

∑ t=1

1 × ̃ Q (υ)

N

∫ ∏ du(s)q(t ) s=1

⎛ ⎡ exp⎜⎜ −β ⎢Hb(α) + ⎝ ⎢⎣

N

⎤⎞

s=1

⎥⎦⎠

∑ W: q(s)⎥⎟⎟

(6)

where β = 1/kT and Q̃ (υ) is the conformational partition function of the chain, corresponding to the Boltzmann weight in eq 6. The entire set of SCF theory equations is given by eqs 5 and 6. The Helmholtz free energy of a homogeneous nematic, FN, can be determined from the standard thermodynamic relationship: FN = UN − TSN. Here, UN and TSN are the average energy and entropy of the nematic. Within the mean-field approximation, these thermodynamic functions read: βUN = nβ⟨Hb(α)⟩ −

nN βυq: q 3

(7)

and ⎡ ⎛n⎞ ⎤ −βTS N = n⎢ln⎜ ⎟ − 1⎥ − n[ln(Q̃ (υ)) + β⟨Hb(α)⟩ ⎝ ⎠ ⎣ V ⎦ + Nβ W: q]

(8)

In eq 8, the first term is the ideal-gas contribution while the remaining part is the conformational entropy of n chains. When βUN and −βTSN are combined to calculate βFN, the contributions from the average bonded energy, n β ⟨Hb(α)⟩ mutually cancel. Nevertheless, they are included into eqs 7 and 8 for completeness. Within the simple mean-field description the free energy of the isotropic phase, FI, is an υ-independent constant, which follows directly from eqs 7 and 8 substituting q = 0. In this case, the conformational partition function, Q̃ o, describes ideal chains. Because FI is υ-independent, it is convenient to refer FN with respect to FI. Without loss of generality, we can assume that the director of the nematic phase is oriented along the zaxis. In this case, q is a diagonal matrix with elements expressed as qxx = qyy = −S/2 and qzz = S. The quantity S is the order parameter, quantifying the strength of nematic ordering. With this choice (considering that the mean field W has the same 753


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Figure 3. POM-S measurements of the IN transition in PFO-33 and P3EHT-181. (a) POM images of PFO-33 in the nematic phase (top image), during the transition (central image) and in the isotropic phase (bottom image). The temperatures at which the images were acquired are included in the images. Scale bars correspond to 250 μm. Panels b and e show the relative transmittance spectra recorded during the phase transitions for PFO33 and P3EHT-181, respectively. A POM image of the nematic phase of P3EHT-181 is displayed in the inset. The evolution of the conversion index (C.I., defined as normalized integral value of the transmitted light between 450 and 750 nm) are plotted against temperature in panels c and f for PFO-33 and P3EHT-181, respectively. Panels d and g depict |d(C.I.)/dT| from which the TIN are determined as the peak temperatures.

symmetry as q), the SCF theory equations and thermodynamic functions can be substantially simplified after straightforward algebraic manipulations. In particular, for the free-energy difference, βΔFN = βFN − βFI, one obtains: ⎡ ⎛ Q̃ (υ) ⎞⎤ N ⎟⎟⎥ β ΔFN(υ) = n⎢ βυS2(υ) − ln⎜⎜ ⎢⎣ 2 ⎝ Q̃ o ⎠⎥⎦

The average in eq 10 is taken over unperturbed WLC conformations (i.e., ideal chains) placed randomly into the converged field. To be accurate, this procedure requires the generation of a large number of conformations.37 At the same time, it must be performed only for one value of υ. For the remaining values, ln(Q̃ (υ)) can be obtained37 by integrating dln(Q̃ (υ))/dυ.

■

(9)

RESULTS AND DISCUSSION Determination of Nematic−Isotropic Transition in PFO and P3EHT. The TIN of our series of PFOs and P3EHT were experimentally determined by means of POM-S. Examples of the data analysis carried out are shown in Figure 3 (for PFO-33 and P3EHT-181). Figure 3a depicts POM images of PFO-33 acquired at different temperatures during the phase transition. The image on the top is taken at 250 °C, when PFO33 is in the nematic phase; the central image is acquired at 259 °C during the nematic−isotropic transition; and, finally, the image on the bottom is acquired once the PFO-33 is in the isotropic phase. It is clear from this sequence of images that during the IN transition the color intensity decreases until the image becomes dark, i.e., no light passes through the sample, indicating that the material is in the isotropic phase. A POM image of the nematic phase of P3EHT-181 is displayed in the inset of Figure 3e. Let us now quantitatively analyze the isotropic−nematic transition of our polymers. For this purpose, we measured the intensity of the transmitted light as the temperature was increased, by means of visible spectroscopy. Figure 3b,e shows the relative transmittance spectra (relative transmission vs wavelength, λ) recorded at different temperatures during the phase transitions for PFO-33 and P3EHT-181 samples,

In eq 9, the notation S(υ) explicitly indicates that the magnitude of the order parameter in the mean field is determined by the strength of the MS constant. To solve the set of SCF equations and to calculate the free energy, we employ an iterative procedure based on the concept of partial enumeration of conformations.66−68 The details of the approach, as implemented for nematic polymers described by a discrete WLC model with MS interactions, have been elaborated elsewhere.37 Here it is mentioned briefly that the iterative procedure improves gradually the estimation of Wzz (the other components follow from symmetry). At the converged state, Wzz complies with the average orientation tensor characterizing the segments of WLC in this field (see eq 5). For each Wzz during the iteration steps, the average q is obtained37 as an average over WLC conformations generated with Monte Carlo sampling according to the Boltzmann weight in eq 6. To obtain the free energy, it is necessary to calculate Q̃ in the converged nematic field. One takes into account37 (cf. eq 6) that Q̃ = Q̃ o

⎛ N ⎞ exp⎜⎜ −∑ β W: q(s)⎟⎟ ⎝ s=1 ⎠

o

(10) 754


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dependence of the MS parameter, υ(T). In actual materials, chain length can affect TIN through additional intermolecular mechanisms, such as changes in local liquid packing (e.g., due to different concentrations of chain ends, inversely proportional to MW). Because SCF is effectively a single-chain theory, effects related to intermolecular correlations cannot be considered explicitly in top-down mapping. They can result into effective MW-dependency of estimated MS parameters. However, we expect that such effects of chain length on υ(T) are secondary, taking also into account the moderate range of MW considered here. The significance of intramolecular effects is supported by early SCF studies64 of nematic polymers using MS models with constant υ. In these studies, the dependency of TIN on chain length has qualitatively similar trends as Figure 4. The method is realized by considering for each melt the difference between the Helmholtz free energies of homogeneous nematic and isotropic phases at TIN(α), as a function of the free parameter υ (cf. eq 9). The point where βΔFN (υ) = 0 corresponds to the required υα(TIN(α)) . The step used to probe υ in the SCF theory calculations corresponds to 0.005 kTIN(α). We remind that for the current incompressible MS model it is legitimate to locate the transition via the Helmholtz free energy. In compressible MS models other thermodynamic potentials, e.g., Gibbs or grand-canonical free energies, should be employed. In Figure 5, we present the results of SCF calculations for υα(TIN(α))/kTIN(α) as a function of Nα, for PFO (black symbols)

respectively. The relative transmission spectra were then integrated between 450 and 750 nm and the resulting value normalized between 1 and 0 to obtain the conversion index, C.I., of the phase transformation. Being a 1-st order transition, the IN transition exhibits sigmoidal curves when plotted against temperature (Figure 3c,f). The absolute values of the derivatives of these sigmoidal curves exhibit a maximum at the temperature at which the phase transformation occurs more rapidly, i.e., at the TIN (Figure 3d,g). Thus, we obtain TIN = 258 °C for PFO-33 and TIN = 111 °C for P3EHT-181. We note that that the validity of our spectroscopic method is demonstrated by comparing with a second technique, differential scanning calorimetry (DSC), as shown in the Supporting Information. The TIN extracted for all PFO samples using the spectroscopic approach are presented in the top panel of Figure 4 as a function of the degree of polymerization of PFO. For

Figure 4. Dependence of temperature of isotropic−nematic transition, TIN, on average number of monomers, Ñ α, in PFO (upper panel) and P3EHT (lower panel) as obtained in experiments.

P3EHT, the analogous plot is presented in the bottom panel of Figure 4. In this case, the TIN for samples other than P3EHT181 were taken from an earlier study.25 Figure 4 illustrates clearly the shift of the IN transition to higher temperatures upon increasing the length of the PFO and P3EHT chains. The saturation trends observed for P3EHT-181 demonstrate that our estimates of TIN are free of serious artifacts due to kinetics of ordering, i.e., they are reasonably close to the true thermodynamic transition. Indeed, we would not have observed saturation behavior if the onset of disorder upon heating were affected by protracted polymer dynamics (especially for the longest P3EHT-181 sample). Temperature Dependence of MS Parameter. The effect of chain length on the IN transition experimentally observed and documented in Figure 4 allows us to estimate the temperature-dependent MS constant as follows. We map on the WLC model PFO and P3EHT chains with polymerization degrees considered in Figure 4. For each of these chain lengths, we fix in the canonical ensemble of the SCF theory the temperature of the system to the temperature of the IN transition TIN(α) (taken from Figure 4) and find the υ at which the IN transition occurs. Because the estimated υ is free from effective MW-dependency related to intramolecular correlations (cf. Introduction), considering υ as a function of TIN allows us to estimate, to leading order, a bare temperature

Figure 5. Main panel: SCF theory results for the magnitude of Maier Saupe constant in units of thermal energy, υα(TIN(α))/kTIN(α), as a function of the number of coarse-grained segments, Nα, for PFO (black symbols) and P3EHT (red symbols). Inset: same as main panel but for a P3EHT-like model with temperature-independent persistence length.

and P3EHT (red symbols). To facilitate the quantitative assessment of the magnitude of the MS constant, υα(TIN(α)) is presented normalized by the thermal energy kTIN(α). The length of the error-bars corresponds to 0.0025 (half the step used to probe υ in SCF calculations) and is about the size of the symbol. One observes that to obtain nematic order for longer chains, the magnitude of the MS constant in units of thermal energy must be increased. This effect is pronounced for PFO but very weak for P3EHT, where for the considered chain lengths υα(TIN(α))/kTIN(α) changes by about 1%. Studies where lp is temperature independent, e.g., refs 34, 36, 38 and refs 63, 755


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Chemistry of Materials 84 (the latter, when considering the rigid-rod limit) predict an opposite effect. Namely, the magnitude of the MS constant (in units of thermal energy) leading to a nematic phase is reduced for longer chains. In studies with constant lp, this effect stems from long-range intramolecular orientational correlations introduced along the chain backbone through the bending rigidity. Due to orientational synergy between intramolecular segments, as the polymer becomes longer, weaker nonbonded interactions are required to bring nematic order. In our case, however, the T-dependence of lp leads to a “negative feedback loop”. Specifically, orientational correlations become weaker for longer chains since they are considered at higher temperatures, TIN(α) where the persistence lengths are smaller. The reduction of long-range intramolecular orientational correlations along the backbone of these more flexible chains is such, that larger values of υα(TIN(α))/kTIN(α) are required to obtain nematic order. This argument is in agreement with the weak dependence of υα(TIN(α))/kTIN(α) on Nα for P3EHT, where the persistence length is substantially less sensitive to temperature, comparing to PFO. Plausibly, in the case of P3EHT-65 the gain from increasing the chain length overrides the counterbalancing effect of reduced persistence length which leads to the slight bend of the curve downward. Regarding this point, quantifying υα(TIN(α))/kTIN(α) in the longest sample P3EHT-181 can bring more clarity. For such long chains, however, accurate SCF calculations with partial enumeration require additional method development and were not considered here. As an additional, simple, validation of these arguments, the inset of Figure 5 presents υα(TIN(α))/kTIN(α) as a function of Nα for P3EHT-like systems, where the persistence length in SCF calculations is fixed at all temperatures to lp = 2.53 nm. As expected, υα(TIN(α))/kTIN(α) decreases with Nα. The choice of the fixed value for lp is rather ad hoc and aims only to illustrate the qualitative differences introduced by assuming a temperature-independent persistence length. Similar trends were observed when fixing the persistence length for PFO to arbitrary value (data not shown). Inspired by phase diagrams33 in SCF studies based on MS models combined with continuum WLC representation of polymer architecture, we consider the phase behavior of our MS model (based on discrete WLC) in terms of two universal parameters Lα/lp(α)(TIN(α)) and υα(TIN(α))Nα/kTIN(α). The first parameter essentially quantifies the flexibility exhibited by the chain on the scale of its contour length, whereas the second parameter compares the magnitude of the nematic energy per chain with the thermal energy. In Figure 6, we replot (open symbols) the boundary of the isotropic and the nematic phases from Figure 5 in terms of these two parameters. For this straightforward transformation, we again use TIN(α) from Figure 4, whereas lp(α)(T) and Lα follow from Modeling section. To illustrate better the universality concept, Figure 6 includes data points (crosses) corresponding to SCF calculations with temperature-independent persistence lengths (see inset of Figure 5). It is instructive36 to add to Figure 6, the location of the isotropic−nematic transition for molecules represented by a rigid-rod model. Here, as representative examples of such materials we consider samples of DEH-PPV homopolymers investigated earlier.32 DEH-PPV is known to be a stiff molecule, where a persistence length on the order31 of 11 nm has been reported85 based on light scattering data. The contour lengths of the DEH-PPV homopolymers investigated in ref32 are smaller or comparable (at most) with this estimate of persistence length. This justifies their consideration as rigid

Figure 6. Results of SCF theory calculations from Figure 5 presented as a universal plot. Open symbols correspond to the actual case of a temperature-dependent persistence length (black and red colors stand for PFO and P3EHT, respectively). Crosses mark the data obtained for a P3EHT-like model postulating a temperature-independent persistence length (cf. inset Figure 5). The arrow and the solid circle mark the prediction for DEH-PPV samples from ref 32 when described as rigid rods.

rods. Of course, the rigid-rod approximation will not hold for long DEH-PPV homopolymers, which will be more coil-like having large Lα/lp(α) ratios. We take this chance to remind that in ref 32 the number of segments used to map DEH-PPV molecules on the rigid-rod model reflects an amount of volumetric units.86,87 Taking into account that rigid rods have an infinite persistence length, for all these DEH-PPV Lα/lp(α) = 0 irrespective of polymerization degree. Moreover, for rigid polymers the mean-field location of the IN transition is given by υα(TIN(α))Nα/kTIN(α) ≃ 4.54. We mention that calculations32 based on the orientation tensor si(s) = 2qi(s)/3, lead to υα(TIN(α)) Nα/kTIN(α) ≃ 6.811 (which contains a 3/2 prefactor, comparing to 4.54). In view of these arguments, all DEH-PPV samples correspond on the universal plot of Figure 6 to a single point (arrow and solid circle). Overall, the plots in Figure 6 comply quite well with the expected universalities in phase behavior. Interestingly, one can observe an approximate equivalence between PFO-28 and P3EHT-31, as well as between PFO-33 and P3EHT-37 samples. The minor deviations from universality observed around the data points describing these samples, are presumably due to the discrete WLC model. In particular, in PFO the same ratio Lα/ lp(α)(TIN(α)) is realized with a WLC containing a smaller (slightly) amount of segments comparing to the WLC representing the matching P3EHT (cf. Tables 1 and 2). Hence, the WLC of PFO has a smaller conformational entropy than the equivalent WLC of P3EHT. To establish the temperature dependence of the MS constants, υα(TIN(α))/kTIN(α) from the main panel of Figure 5 are replotted in the main panel of Figure 7, as a function of 1/ TIN(α). Interestingly, υα(TIN(α))/kTIN(α) increases as the temperature grows. This behavior contrasts previous studies,25,36 where υα(TIN(α))/kTIN(α) was found to decrease increasing temperature. Assuming a simple linear dependence: 756


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the linear approximation, the positive slope of the plot in the inset corresponds to a MS parameter with positive A term in agreement with refs 25, 36. Considering that the plots in the main panel and inset of Figure 7 are based on the same lengths of P3EHT molecules, let us suppose that our MS parameter contains effective (secondary) dependencies on MW, i.e., that we actually have estimated υP3EHT(TIN(P3EHT), NP3EHT) . In this case, the plots in Figure 7 would mean that υP3EHT(TIN(P3EHT), N P 3 E H T ) f o r T - d e p e n d e n t l p ( P 3 E H T ) d iff e r s f r o m υP3EHT(TIN(P3EHT), NP3EHT) for constant lp(P3EHT). Therefore, effective MW-dependencies of υP3EHT will not change our main conclusion that assumptions made on the temperature dependence of persistence length affect qualitatively the Tdependence of the MS parameter obtained from top-down mapping. If we assume that the anomalous Bα term has indeed an entropic origin, the signs of Aα and Bα discussed in connection with Figure 7 imply that entropic contributions to our MS constants favor the formation of the nematic phase. This scenario does not contradict basic physics and can be rationalized through a simple, qualitative, argument. When groups of monomers are brought close to each other, the allowed conformations of side chains and backbone rearrangements are restricted by steric, athermal, exclusions. On a coarsegrained level, the interactions between two such groups (denoted in the following 1 and 2) are represented by an effective local potential U(r2 − r1) = Δfδ(r2 − r1) . Here Δf = f − f 0, where f and f 0 are the free energies of the pair, when the two groups are close to and far from each other, respectively. Within the simple overlap approximation92 Δf = kTV0 where V0 is the overlap volume of the two groups. If u1 and u2 are vectors defining the orientation of the coarse-grained segments representing these two groups, we can estimate V0 as an excluded volume ν|u1 × u2|, where ν is an athermal prefactor.93 To estimate the excluded volume of two segments as a function of their orientation we employed the well-known result of the Onsager theory.48 Expanding49,94 |u1 × u2| in spherical harmonics delivers U(r) ≃ kT ν δ(r2 − r1) (c1 − c2 q1:q2), where c1 and c2 are known (positive) arithmetic coefficients.49,94 The anisotropic part in this approximation delivers a Maier−Saupe potential with a prefactor of purely entropic nature: − kTνc2δ(r2 − r1) q1:q2.

Figure 7. Main panel: SCF theory results for the magnitude of Maier Saupe constant in units of thermal energy, υα(TIN(α))/kTIN(α), as a function of inverse temperature, 1/TIN(α), for PFO (black symbols) and P3EHT (red symbols). Dashed lines are linear fits to these data. Inset: same as main panel but for a P3EHT-like model with temperature-independent persistence length.

υα(T ) A = α + Bα kT T

(11)

we perform linear regression of data in Figure 7 for PFO and P3EHT (dashed black and red lines, respectively). This linear approximation is analogous32 to the simple empirical form frequently used to describe the temperature dependence of the Flory−Huggins (FH) parameter.88−91 The quantities Aα and Bα are temperature-independent constants.25,32 For the FH parameter it is a common practice91 to name them as “enthalpic” and “entropic” contributions, respectively. However, these names can be misleading concerning the actual physical mechanisms underlying these terms.88 Hence, we use here for Bα an alternative designation, “anomalous” term. The regression delivers APFO/k = −535.9 K and BPFO = 3.1916, whereas AP3EHT = −169.39 K and BP3EHT = 2.4669. Within this simple analogy to the FH parameter, we observe that the nematic order is promoted by the anomalous B term, whereas the A term actually destabilizes the nematic phase (cf. eq 3 where βHnb(α) ∼ −υα(T)). In previous studies,25,36 the linear approximation leads to opposite trends, i.e., positive A and negative B terms (as a consequence of the different monotonicity of υα(TIN(α))/kTIN(α)). It is emphasized that the different results on how υα(TIN(α))/ kTIN(α) changes with temperature, stem from the temperatureindependent persistent lengths considered in the previous studies.25,36 To illustrate this point, the inset of Figure 7 presents υα(TIN(α))/kTIN(α) as a function of 1/TIN(α) obtained from SCF calculations for P3EHT-like systems with fixed persistence length (cf. inset of Figure 5). In this case, υα(TIN(α))/kTIN(α) decreases increasing temperature, as in refs.25,36 The difference in how υα(TIN(α))/kTIN(α) changes with 1/TIN(α) in cases of temperature-dependent and temperature-independent persistence lengths, manifests the phenomenon already described in context of Figure 5. Due to noticeable deviations from linear behavior, we do not attempt a linear regression of the data obtained for the hypothetical case of fixed lp(α). It is worth mentioning, nevertheless, that within

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SUMMARY AND CONCLUSIONS

Our study was motivated by the interest in parametrizing MS models of nematic conjugated polymers using experimental data. We focused mainly on two materials, PFO and P3EHT, corresponding to semiflexible and flexible polymers, respectively. For their theoretical description, a variant of the MS model was employed, representing these two polymers by discrete worm-like chains with temperature-dependent persistence length. The model assumes the polymer liquid to be incompressible and the MS potential accounts phenomenologically for the combined effect of various microscopic factors promoting collective chain alignment in the actual materials. For P3EHT, the simple temperature dependence of the persistence length realized by our model is comparable to trends reported experimentally61 for P3HT (belonging to the same family of polymers as P3EHT). For PFO, to the best of our knowledge, such experimental data are currently not available. We extended the spectrum of addressed molecular flexibilities considering also a rod-like polymer (infinite 757


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Chemistry of Materials persistence length), benefiting from older data32 for short DEH-PPV homopolymers. The work focused on estimating the magnitude and the temperature dependence of the MS parameter, υ(T). For this purpose, we followed an established strategy,14,25,32 utilizing experimental data on the temperature of the IN transition. These temperatures were obtained for PFO and P3EHT samples with several molecular weights, through polarized optical microscopy and spectroscopy. As expected from previous studies,14,25,32,62 the transition temperatures increased with increasing chain length. For the experimentally investigated samples, the theoretical description of the IN transition was obtained considering the thermodynamics of our model within a numerical SCF theory based on partial enumeration. The magnitude of υ(T) was obtained by matching the temperature of the IN transition for different samples in the model and experiments. Due to the shift of the IN transition with increasing chain length, the approach delivers values of υ(T) at different transition temperatures. Our central conclusion is that the T-dependence of the MS parameter is significantly affected by whether or not the model of polymer architecture accounts for changes of persistence length with temperature. For WLC models where increasing temperature leads to higher chain flexibility, we observed that υ(T)/kT is an increasing function of T. In contrast, in test cases where we fixed the persistence length of the WLC, υ(T)/kT was found to be a decreasing function of T, which agrees with previous studies based on rod-like descriptions25,32 or a WLC model36 with a T-independent persistence length. For Tdependent persistence lengths, approximating the changes of υ(T)/kT with temperature as υ(T)/kT = A/T + B lead to a picture where nematic order is favored by the anomalous B term but disfavored by the A term. Conversely, the changes of υ(T)/kT with temperature for fixed persistence lengths, correspond to a case where nematic order is promoted by the A term. If one assumes that A and B have indeed enthalpic and entropic origins, the behavior observed for T-dependent persistence lengths corresponds to a situation where the nematic order is favored by entropy but disfavored by enthalpy. In principle, obtaining nematic order through entropic contributions to the MS parameter is not paradoxical. It can be rationalized through effective anisotropic steric repulsions due to reduction of allowed microstates when two conjugated polymers are close to each other. We believe that our results do not contradict the studies just mentioned25,32,36 but highlight the strong phenomenological character of the MS constant estimated from top-down approaches. Specifically, MS constants extracted by mapping flexible polymers such as P3EHT25 on rod-like descriptions (infinite persistence length), cannot be directly compared with MS constants estimated from WLC models. In the former, the MS constant encapsulates effects of temperature on chain flexibility. In the latter, such effects are excluded from the MS constant due to explicit description of chain architecture. The implementation of WLC with fixed persistence length in ref36 was justified by atomistic simulations of P3HT at T ≥ 600 K, which is well above the temperatures addressed in our study. Combining ref 36 and our results suggests that υ(T)/kT (a least in poly(alkylthiophenes)) may have nonmonotonous temperature dependence, i.e., presents a decreasing function of T at high temperatures36 whereas being an increasing function at lower temperatures (our case). Within our approach, potentially observing such behavior (for example in P3EHT)

requires further studies based on long polymers. Such a nonmonotonicity would indicate that the simple approximation υ(T)/kT = A/T + B is not applicable across a broad range of temperatures and more elaborated forms should be explored. In view of the phenomenological nature of the MS parameter, we also expect that the values of the coefficients A and B for the materials considered in this study, will depend on the details of the polymer model, such as the implementation of a discrete or continuum WLC. From an experimental point of view, our results highlight the importance of having conjugated polymers extensively characterized with respect to persistence length and its dependence on various control parameters.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.7b04194. Measuring molecular weight of PFO samples, details on estimating the temperature of the isotropic−nematic transition via polarized optical microscopy and spectroscopy, determination of the isotropic−nematic transition in PFO using differential scanning calorimetry, determination of the refractive index increment (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*J. Martin. E-mail: [email protected]. *R. A. Segalman. E-mail: [email protected]. *N. Stingelin. E-mail: [email protected]. *K. Ch. Daoulas. E-mail: [email protected]. ORCID

Jaime Martin: 0000-0002-9669-7273 Rachel A. Segalman: 0000-0002-4292-5103 Natalie Stingelin: 0000-0002-1414-4545 Kostas Ch. Daoulas: 0000-0001-9278-6036 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Marcus Müller and Kurt Kremer for many helpful discussions related to this work and the useful comments made after reading our manuscript. J.M. acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant, agreement no. 654682, the Provincial Council of Gipuzkoa under the programme Fellow Gipuzkoa. R.A.S. gratefully acknowledges support from the NSF-DMR Polymers Program through grant no. 1608297. C.G. and K.C.D. acknowledge support from the German Federal Ministry for Education and Research (BMBF) within the POESIE project (FKZ 13N13694).

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REFERENCES

(1) Patel, S.; Chabinyc, M. Anisotropies and the Thermoelectric Properties of Semiconducting Polymers. J. Appl. Polym. Sci. 2017, 134, 44403. (2) Street, R. Unraveling Charge Transport in Conjugated Polymers. Science 2013, 341, 1072−1073. (3) Himmelberger, S.; Salleo, A. Engineering Semiconducting Polymers for Efficient Charge Transport. MRS Commun. 2015, 5, 383−395.

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