Assessing the influence of nanoscale morphology on the mechanical properties of semiconducting polymers

Abstract

The ease of processability of conjugated organic polymers, alongside their capability of transporting charges, makes them excellent candidates for applications in flexible and biocompatible electronic devices. In such applications, retaining the electronic properties upon repeated cycles of mechanical strain is key to avoid losing device performance over time. To achieve an accurate mechanical characterization at the nanoscale of these partially crystalline systems, it is critical to have access to reference values of polymer elastic constants and to be able to relate them to the local morphology. With this objective, in the following, we set up a computational protocol for the calculation of elastic constants through molecular dynamics (MD) simulations in the linear deformation regime. We apply such a scheme to the prediction of the elastic behavior of two well-known semiconducting polymers (C16-IDTBT and C14-PBTTT) in crystalline and amorphous phases, showing that the local fluctuations of the Young's modulus can span two orders of magnitude owing to its strong dependence on morphology, anisotropy, and strain direction. The comparison with experimental measurements of the Young's modulus on the nanoscale suggests good agreement in calculated trends.

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Introduction

Semiconducting polymers (SPs) are widely employed as active material in a variety of optoelectronic devices, such as organic field-effect transistors (OFETs),^1–4 organic light-emitting diodes (OLEDs),^5–7 and photovoltaic (PV) cells.^8–11 The underlying reason for their massive application in organic electronics comes from a perfect match between their key structural features (provided by the appealing properties of plastics such as low cost and flexibility) and their optoelectronic properties (traditionally a prerogative of inorganic materials). In particular, semiconducting polymers based on the alternate “donor–acceptor” motif (D–A)^12–14 have emerged as a valid alternative to molecular semiconductors for transistors and solar cells. These copolymers are characterized by larger charge carrier mobility and smaller transport activation energy if compared to homopolymers such P3HT, an observation that points out to a lower intrinsic energetic disorder for the former class of materials.¹⁵ These high-performing SPs are characterized by a spatially extended π-conjugated backbone, which allows multiple interchain contacts and an efficient charge transport even in the amorphous phase, in which short-range intermolecular aggregation¹⁶ and the rigidity of the backbone¹⁷ are sufficient to facilitate long-range charge transport. A famous member of the D–A family is PBTTT (poly(2,5-bis(3-alkylthiophen-2-yl) thieno-[3,2-b]thiophene)),¹⁸ in which the combination of a centrosymmetric bi-thiophene unit substituted with long linear alkyl side chains and unsubstituted thieno-[3,2-b]thiophene central ring creates free volume to promote side-chain interdigitation between adjacent backbones orthogonal to the π-stacking direction. The side chain interdigitation involves the hierarchical arrangement into 3D ordered structures in which π-stacked lamellar sheets register in the out-of-plane direction, promoting large crystalline domains.¹⁹

A complete change of paradigm was triggered by the synthesis of the IDTBT (indacenodithiophene-co-benzothiadiazo) copolymer, which showed high hole mobilities²⁰ without the need for high-crystalline packing. Indeed, XRD patterns recorded for the IDTBT revealed only weak and broad peaks,²¹ consistent with a nearly amorphous microstructure. For IDTBT, the design strategy is based on the reduction of conformational disorder by using extended aromatic structures together with bridging units. This results in a long persistence length for the polymer chains, allowing for efficient intrachain charge transport despite a poorly ordered microstructure. As demonstrated, small ordered domains²² or a few close contact points between polymer chains^23,24 are sufficient to avoid charge trapping at chain ends or due to conjugation breaks.

Despite the versatility of chemical design in improving the electrical performance of D–A polymers, it is only recently that some effort has been put into designing conjugated polymers with desired mechanical properties, a key point in the development of concurrently flexible and durable micromechanical devices.^25–29 To contribute to this endeavor we selected PBTTT and IDTBT as our testbed, and by taking advantage of recent nanomechanical measurements,^30,31 we set up a framework for the theoretical prediction of the elastic constants of amorphous and crystalline polymeric domains. Indeed, a high-resolution description of the mechanical properties is essential due to the micro- and nano-scale size of polymeric SPs film domains, and is nowadays possible to obtain by means of sophisticated atomic force microscopy measurements.³² However, these 2D maps of the elastic constants, albeit useful to overall identify the morphology at the nanoscale and the local response to external stresses, are not sufficient to establish a clear link between the mechanical properties and the morphology down to the molecular scale. Some efforts in this direction have been pursued in the last decade by means of molecular simulations,³³ though mainly focused on amorphous phases and blends of P3HT. In particular, Tummala et al. studied the mechanical fracture of P3HT/fullerene blends at atomistic level,³⁴ and evaluated the tensile modulus of amorphous P3HT³⁵ as a function of the chain length, also highlighting the importance, in these calculations, of modeling long enough chains for describing correctly the entanglement. Also, coarse-grained models were used for simulating the elastic and plastic responses for the same systems;^36–38 interestingly, Giuntoli and coworkers were able to simulate the nanoindentation experiment and compare the results with the simpler tensile test.³⁸ Another important contribution was published by Lipomi and coworkers, which calculated the elastic modulus of three semiconducting polymers in different morphologies and showed that, in determining the final Young's modulus of a neat polymer sample, the variations in morphology are more important than the chemical structure itself. Here we wish to complement these results and focus instead on the determination of the anisotropic elastic response of different microstructures with a controlled and variable degree of structural order.

Force field parametrization and preparation of initial morphologies

We investigated four different samples of IDTBT and PBTTT: one amorphous (a-IDTBT and a-PBTTT), two perfectly crystalline polymorphs with interdigitated alkyl side chains (X1-IDTBT, X1-PBTTT and X2-IDTBT, X2-PBTTT), and one extra crystalline polymorph in which the degree of alkyl side chain interdigitation and order was strongly reduced (labeled as X2d-IDTBT and X2d-PBTTT to differentiate them from the corresponding interdigitated polymorphs), whose structures are shown in the right panels of Fig. 1. All MD simulations on the aforementioned morphologies were run using the NAnoscale Molecular Dynamics (NAMD) software³⁹ with the Dreiding force field.⁴⁰ In the present work, we adopted the force fields of PBTTT and IDTBT developed in ref. 17 and 42 utilizing them without modification. In these studies, the torsional profiles of the bonded aromatic thiophene and thienothiophene rings of the core and the dihedrals between conjugated subunits and alkyl chains were reparametrized against DFT calculations using the B3LYP functional and the cc-pVTZ basis set. Atomic charges were obtained by fitting the electrostatic potential (ESP charges⁴¹) calculated on an oligomer of two repeating units using the same level of theory. The van der Waals σ parameter of hydrogen atoms of IDTBT has been set to 2.5 Å. The force fields were validated by comparing the simulated crystalline structures, obtained through molecular mechanics (MM) and MD simulations, with experimental XRD data. Overall, the force fields showed excellent agreement with the experiments in terms of inter-lamellae and π-stacking distances for PBTTT and monomer unit length for IDTBT, respectively.

Fig. 1 Radial distribution functions g(r) computed between the sulfur atoms for the unperturbed systems (ε = 0%) of (a) IDTBT and (b) PBTTT for their amorphous and crystalline phases. Different views of the π-conjugation between two monomers of (c) X1- and X2-IDTBT, (d) X1- and X2-PBTTT, and (e) X2d-IDTBT and X2d-PBTTT taken from NVT trajectories at T = 298 K. The vertical panels on the right of (c)–(e) highlight the side chain ordering of the aforementioned polymorphs.

Three-dimensional periodic boundary conditions (PBC) were applied to minimize finite-size effects, and the electrostatics were calculated by employing the smooth particle mesh Ewald (PME) method.⁴³ Amorphous samples consisted of twenty-four chains, where each chain was made of 12 and 10 monomers for PBTTT and IDTBT, respectively, so as to obtain an average length of ∼160 Å for the extended chain. The procedure used to generate amorphous phases is the following:

(i) 24 oligomers were randomly inserted in a very large unit cell (300 Å × 300 Å) and subjected to a 500 ps MD run at high temperature (NVT, T = 1000 K) while keeping the density low (∼0.02 g cm⁻³) to favor a random spatial distribution of the oligomers;

(ii) five successive 500 ps-long MD runs (NPT, P = 1 atm, T = 300 K) were performed at decreasing temperatures (1000 K, 500 K, 400 K, 350 K, 300 K);

(iii) samples were equilibrated for 35 ns (NPT, T = 298 K and P = 1 atm, using Berendsen barostat⁴⁴) allowing anisotropic expansion of the orthorhombic cell.

(iv) Box average sizes, corresponding to the lattice dimensions that are stable at atmospheric pressure, were computed from the last 25 ns of the same trajectory.

All crystalline supercells of the highly crystalline polymorphs, namely X1 and X2 of IDTBT and PBTTT, were obtained by using the same approach: starting from previously equilibrated unit cells, consisting of two monomers for PBTTT¹⁷ and one monomer for IDTBT,⁴² we replicated them 10 × 5 × 5 and 10 × 10 × 5 times along the three lattice parameters without affecting the original triclinic shape. To be consistent with the amorphous samples, we did not vary the chain length in terms of constituent monomer units, for a total of 200 polymeric chains. These initially infinite chains were transformed into 12- and 10-mers (for PBTTT and IDTBT, respectively) by randomly removing one monomer–monomer bond in each chain. To further investigate how the mechanical properties evolve while selectively increasing the degree of structural order, we simulated two additional phases labeled X2d-IDTBT and X2d-PBTTT, characterized by a similar π-stacking with respect to X2 polymorphs, but with a reduced degree of side chain interdigitation due to the increased distance between lamellae. The procedure used to generate the non-interdigitated phase is the following:¹⁷

(i) starting from crystalline cells made of three layers of eight π-stacked decamers [dodecamers] of IDTBT [PBTTT], the interlayer distance was increased up to 50 Å, so that the alkyl chains of successive layers are not interdigitated anymore;

(ii) after geometry optimization, the systems underwent a short (50 ps) MD simulation at high temperature (NPT, P = 1 atm, T = 500 K) to induce disorder along and between the polymer chains;

(iii) a 500 ps-long MD simulation was then performed at room temperature (NPT, P = 1 atm, T = 300 K) to let the systems relax.

Structural organization of the polymer chains in absence of constraints

We investigated two crystalline polymorphs of each semiconducting polymer: the X1- and X2-polymorphs of IDTBT and the X1- and X2-polymorphs of PBTTT (Fig. 1c and d). All crystalline unit cells exhibit a triclinic lattice (see Table 1) and are characterized by interdigitated, nearly all-trans aliphatic side chains, C-14 and C-16 for PBTTT and IDTBT, respectively. More precisely, X1-IDTBT exhibits a strong overlap of the aromatic moieties of monomeric units, with a measured shift along the backbone of only 2.5 Å (see Fig. 1c). The π-stacking distance, d_π-stacking, is calculated to be 4.1 Å, and it is computed as d_π-stacking = 〈d sin θ〉_t, where d represents the distances to the nearest chains, corresponding to the first peak of the radial distribution function of Fig. 1a, and θ is the tilt angle between adjacent chains, i.e., the angle opposite to d_π-stacking. As indicated by the subscript, all values are averaged over the simulation time. On the other hand, X2-IDTBT is distinguished by a larger shift of 7.0 Å between the aromatic cores of crystalline planes (Fig. 1c), thus favoring a crossed π-stacking interaction (with a d_π-stacking = 4.3 Å computed as described above) between the IDT and BT moieties belonging to adjacent polymeric chains. Additionally, the two side chains of the monomer are considerably tilted compared to the plane in which the backbone lies, with an angle of almost 45°. The same π-stacking distance of 4.0 Å was estimated for the two PBTTT polymorphs, but a different superposition of the conjugated core, which is higher in X1-PBTTT (8.0 Å) compared to the X2-PBTTT (5.0 Å), see Fig. 1d. Out-of-plane full-trans side chains characterize both PBTTT polymorphs with a tilted angle that differs by less than 5° for the X1-PBTTT with respect to the X2-PBTTT (50.3° and 54.4°, for X1-PBTTT and X2-PBTTT, respectively).

Table 1 Average crystal unit cell parameters, unit cell volumes (V) and densities (ρ) of the samples for the X1-, X2- and X2d-polymorphs of IDTBT and PBTTT obtained after the NPT (T = 298 K and P = 1) equilibration of the sample box

	a (Å)	b (Å)	c (Å)	α (°)	β (°)	γ (°)	V (Å^3)	ρ (g cm^−3)
X1-IDTBT	30.7	7.9	32.1	83.2	114.0	112.2	6579.2	1.10
X2-IDTBT	32.3	8.1	32.0	31.0	56.7	72.6	3305.4	1.12
X1-PBTTT	6.2	20.2	13.8	103.1	146.8	80.7	920.5	1.11
X2-PBTTT	6.1	21.3	13.6	98.5	144.8	83.4	1007.3	1.11
X2d-IDTBT	28.6	15.0	16.0	33.1	41.5	65.9	1976.8	0.86
X2d-PBTTT	6.2	24.8	13.4	98.8	143.5	86.1	1205.8	0.89

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Results and discussion

Calculation of the elastic constants

The mechanical response of the different samples was obtained within the elastic (or linear) regime of deformation. Using Voigt's notation and considering only normal stresses σ and strains ε, the constitutive equation can be reduced to a simple matrix product in which the 3 × 3 symmetric C matrix is the stiffness tensor, and x, y, z are the cartesian axes:^45,46

In practice, the components of the stress tensor upon the application of a given strain are obtained through the following sequence of MD simulations (Fig. 2 shows an example of one crystalline, X2-IDTBT, and one amorphous sample, a-IDTBT, before the application of strain):

Fig. 2 Snapshots taken from NVT trajectory of the X2-IDTBT crystalline (left) and amorphous a-IDTBT samples (right) at T = 298 K. Strain deformations were always applied along a, b and c, even if in triclinic cells b and c cell axes are not parallel to axes y and z.

(i) for the unstrained samples, we ran 35 ns NPT simulations (T = 298 K and P = 1 atm) to further equilibrate the system and compute the average box dimensions over the last 25 ns;

(ii) we fixed the box dimensions to the calculated average, and from a new 50 ns-long NVT simulation (T = 298 K) we measured the average pressure tensor P;

(iii) we applied uniaxial strains (ε) of ±0.5%, ±1% by modifying the box dimensions along the a, b and c cell axes, and rescaling accordingly the intermolecular positions;

(iv) we ran a 10 ns trajectory to equilibrate the sample after the rescaling and we registered the average pressure tensor P′ on a 50 ns production NVT simulation at T = 298 K;

(v) we derived the stress tensor at a certain applied strain as the difference σ = −(P′ − P).

For long enough simulations, the off-diagonal components of the stress tensor are vanishing, while the diagonal ones can be inserted in eqn (1) to infer the values of the stiffness matrix.

In practice, the determination of the stiffness tensor C from the MD simulation trajectories was accomplished separately for each sample with an in-house Python code, performing a global fit over all the applied strains and minimizing the difference between the stress tensors actually measured from the simulations and those obtained by using the fitted values of C and eqn (1). From the stiffness matrix, we then computed the compliance tensor as S = C⁻¹ whose elements, if one neglects the presence off-diagonal elements coupling shear and normal stresses in the full 6 × 6 stiffness tensor, can be expressed in terms of both the Young's moduli (E_α) and the Poisson's ratios (ν_αβ), with α, β = x, y, z as shown in eqn (2):

Although the same methodology for calculating the components of the Young's modulus and the Poisson's ratio tensors was implemented for all the considered phases, the anisotropy of the medium has an impact on the number of independent terms of C. Amorphous phases can be reasonably accounted as isotropic media, which conveys C to be described only by two independent terms, i.e. a diagonal and an off-diagonal one. Therefore, the stiffness matrix is completely determined by only two isotropic elastic constants E_ISO and ν_ISO. On the other hand, the description of the mechanical properties of the crystalline structures necessarily involves the anisotropy of the system: C is defined by six independent terms, and consequently, three Young's moduli and six Poisson's ratios were calculated.

Elastic properties

As described in the previous section, all elastic constants were determined from the average pressure tensor, obtained from independent simulations in which one of the three cell axes was selectively deformed by ±0.5%, ±1%, and are reported in Tables 2–4. Owing to the triclinic symmetry of the crystal cells of the X1- and X2-polymorphs of IDTBT and the X1- and X2-polymorphs of PBTTT (see Table 1), the three Young's moduli E_x, E_y and E_z can be very different from each other. In both polymers, the largest Young's modulus was found along the backbone direction (the c axis, which has its largest component along z), essentially because this deformation corresponds to a stretching or a compression of covalent bonds. The second largest Young's modulus component, either E_x or E_y, approximately corresponds to the π-stacking direction, which is found along a for IDTBT, b for PBTTT. Finally, the lowest Young's modulus is associated with the side alkyl chain direction. To compare the overall stiffness of these polymers, it is more straightforward to evaluate the isotropic Young's modulus E_ISO = (E_x + E_y + E_z)/3 and the Voigt bulk modulus B_Voigt (, where C_xx, C_yy, C_zz and C_xy, C_yz, C_xz are the diagonal and off-diagonal elements of C, respectively). The E_ISO of IDTBT was found to be half of that of PBTTT, which is consistent with the increase of the alkyl chains for the former polymer (C16 with respect to C14^47,48), whereas crystalline polymorphs of the same polymer have, as expected, a very similar stiffness behavior. Minor differences were found for the B_Voigt, although the higher values computed for the PBTTT polymer are consistent with the larger Young's moduli. Overall, values are notably smaller if compared to those of Kevlar, a well-known stiff polymer, and larger than those measured for PBT (poly-butyleneterephthalate) and PP (poly-propylene) polymer composites, likely due to the presence of “soft” alkyl side chains attached to the conjugated polymer backbones.

Table 2 Young's moduli (E_x, E_y, E_z and E_ISO) and Voigt bulk modulus (B_Voigt) for IDTBT and PBTTT crystalline polymorphs computed from MD simulations at T = 298 K. The isotropic experimental Young's modulus (E_ISO) are taken from ref. 49 for Kevlar and from ref. 50 for PBT (poly-butyleneterephthalate) and PP (poly-propylene) polymer composites

	E x (GPa)	E y (GPa)	E z (GPa)	E ISO (GPa)	B Voigt (GPa)
Kevlar	—	—	—	130–185	—
PBT + 30% glass fiber	—	—	—	10	—
PP + 40% glass fiber	—	—	—	6.8–7.2	—
X1-IDTBT	6	17	46	23	19
X2-IDTBT	15	25	35	25	17
X2d-IDTBT	10	21	32	21	15
X1-PBTTT	35	13	76	41	21
X2-PBTTT	21	11	88	40	23
X2d-PBTTT	17	5	86	36	20

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Table 3 Poisson's ratios for IDTBT and PBTTT crystalline polymorphs computed from NVT simulations at T = 298 K

	ν xy	ν xz	ν yx	ν yz	ν zx	ν zy
X1-IDTBT	0.43	0.07	0.26	0.42	0.02	0.30
X2-IDTBT	0.55	0.32	0.67	0.58	0.06	0.09
X2d-IDTBT	0.44	0.52	0.57	0.53	0.10	0.11
X1-PBTTT	0.10	0.43	0.08	0.28	0.37	0.04
X2-PBTTT	0.12	0.53	0.07	0.39	0.52	0.06
X2d-PBTTT	0.22	0.47	0.10	0.33	0.40	0.20

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Table 4 Isotropic elastic constants for PBTTT, IDTBT, and some common polymers reported for comparison. “MD” labeled values are obtained from MD simulations at T = 298 K

	ν ISO–MD	ν ISO–EXP	E ISO–MD (GPa)	E ISO–EXP (GPa)	ρ (g cm^−3)	|R⃑| (Å)
Experimental values (labeled as “EXP”) are taken from:a Standard experimental tables.^52b Buckling measurements on a thermally annealed semicrystalline thin films.^51c Film-on-water experiments on semicrystalline thin films.^21 Amorphous phase densities (ρ) and end-to-end distances, |R⃑|, with their associated standard deviations, are reported for IDTBT and PBTTT only, and they are computed from NPT equilibrated trajectories. Plots reporting the end-to-end distribution over time, per polymeric chain, are reported in Fig. S29 and S30 of the ESI.
PS	—	0.35	—	3.4^a	—	—
PMMA	—	0.36	—	3.0^a	—	—
Rubber	—	0.50	—	0.05^a	—	—
a-PBTTT	0.42	—	1.7	1.8^b	1.06	45 ± 18
a-IDTBT	0.42	—	1.4	0.7^c	1.05	67 ± 24

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As S in eqn (2) is defined both in terms of Young's moduli and Poisson's ratios, the fitting procedure that we developed allowed us to compute these additional elastic constants. The determined Poisson's ratios, reported in Table 3, are all positive and sometimes larger than 0.5, the maximum value for an isotropic material. Less straightforward is to relate them to the arrangement of the polymeric chains inside the crystal cell.

We next investigated the mechanical response within the X2d-IDTBT and X2d-PBTTT polymorphs, whose structures are shown in Fig. 1e, for which we introduced conformational and positional disorder within the side chains from the highly ordered crystalline polymorph of X2-IDTBT and X2-PBTTT (cf.Table 1). We observed a similar shift between the monomer aromatic backbones in X2d-IDTBT and X2d-PBTTT, measuring 6.5 Å and 6.6 Å, respectively. In contrast, the original overlap of the crystalline structures was approximately 7 Å for X2-IDTBT and 5 Å for X2-PBTTT. A similar d_π-stacking was also computed for the X2d-polymorphs, corresponding to 4.1 and 3.9 Å for IDTBT and PBTTT, respectively. The loss of alkyl chain ordering affects the computed elastic constants, which are characterized by an overall decrease, compared to the highly ordered phase, of the Young's moduli along all the cartesian directions. Not surprisingly, the most significant drop in the stiffness of the polymeric structure was detected in E_x for X2d-IDTBT and in E_y for X2d-PBTTT, which are mostly representative of the alkyl side chain spacing, i.e. the distance between lamellae, that is along a and b for IDTBT and PBTTT, respectively (see Table 2). For the Voigt bulk modulus, we recorded a drop in magnitude, compared to the corresponding crystalline interdigitated phases, for the sole IDTBT, while the same value was derived for both X2-PBTTT and X2d-PBTTT. We can attribute this contrasting behavior to the reduced extent of the lateral side chains of PBTTT, where the degree of interdigitation is less significant. The effect of disordering on Poisson's ratios is different, and points towards a reduction of the material anisotropy: specifically, ratios that were large in the unperturbed original structure decrease, while those that were small increase.

In order to complete the picture of the mechanical behavior, we also computed the elastic constants for the amorphous samples. Radial distribution functions, g(r), are shown in Fig. 1a and b: extremely broad peaks characterize the amorphous phase, where the most intense peak is the one corresponding to the intermonomer distance along the backbone. As reported in Table 4, the same values of Poisson's ratio ν_ISO were quantified for the two semiconducting polymers, which is within the range reported for common polymeric materials, that goes roughly from a value of 0.35 for polystyrene to 0.5 of natural rubber. Regarding the isotropic Young's modulus E_ISO, we predict that the amorphous phase of PBTTT is slightly stiffer than that of IDTBT, with a discrepancy of ∼0.3 GPa. We ascribed this variation to the larger density and increased length of flexible alkyl side chains (C16 instead of C14 for IDTBT and PBTTT, respectively) incorporated into the monomer's chemical structure. The shorter end-to-end distance, |R⃑|, observed for PBTTT compared to IDTBT, reported in Table 4, indicates the presence of stronger short-range intermolecular interactions in PBTTT. This is supported by the emergence of a peak below 4 Å in the g(r) of PBTTT, which is absent in the corresponding g(r) plot for IDTBT, where the first peak appears at distances greater than 4 Å, see Fig. 1a and b. Additionally, the larger number of torsions per unit length for PBTTT as compared to IDTBT likely contributes to the reduced end-to-end distance. Structural insights from Fig. S1 and S2 of the ESI† further support this latter interpretation: in PBTTT, the rotation of the terminal thiophene unit results in significant deviations from intermonomer bond parallelism. In contrast, this effect is slightly reduced in IDTBT, where a 180° rotation of the IDT unit relative to the BT unit, partially preserves end-to-end molecular alignment.

Structural parameters and their modifications under strain

This section focuses on the structural modification occurring when perturbing the unstretched amorphous and crystalline samples. We calculated first the orientational order parameters 〈P₂〉_b and 〈P₂〉_s, namely the one along the aromatic backbone and that along the end-to-end vector for side chains, respectively, both at rest and after the application of uniaxial strains (values are reported in Tables S1 and in S2 for IDTBT and PBTTT, respectively, of the ESI†). For the crystalline phases, as could be expected from a highly ordered structure, 〈P₂〉_b values are approaching the maximum possible value of 1 with a very low standard deviation. It is interesting to notice that for the X2d-IDTBT, X2-PBTTT crystalline structures, the orientational order is moderately reduced compared to the corresponding interdigitated polymorph. The 〈P₂〉_s values do not reflect the absolute side-chains orientational ordering as they are calculated with respect to phase director n of 〈P₂〉_b, parallel to the backbones. Thus, the almost zero values of 〈P₂〉_s do not indicate a lack of orientational order. They provide, instead, insights into the relative orientation of lateral chains with respect to the backbone direction of propagation. Low, near-zero values, if associated with a low standard deviation, refer to a tilt angle that is very close to the magic angle of 54.47°, as for the case of the interdigitated polymorphs. On the contrary, low 〈P₂〉_s with large standard deviations, like in the case of the side-chain disordered crystalline samples, are representative of a nearly isotropic distribution of orientations. Actually, the application of uniaxial strains has an extremely low effect on both side chain and backbone orientation of crystalline samples, though in expansion the orientational order of the backbone slightly increases and that of the side chains becomes more negative, while as expected, the amorphous samples have low 〈P₂〉_b values and do not show specific trends upon deformation.

We also investigated the impact of strain effects on torsion angles along the polymer backbones. Indeed, previous studies^53,54 have demonstrated that a highly twisted backbone is significantly detrimental to charge transport. Therefore, we calculated the deviation from planarity of the π-core dihedrals in a range of strain ε = −1% to +1%, with a step of 0.5%. Since we applied only small amounts of strain, to ensure not exceeding the elastic region, we observed correspondingly small variations in the deviation from planarity compared to the values measured in the undeformed samples. This is also supported by the plots of dihedral distribution versus strain reported in the ESI† (Fig. S1–S3). To highlight these variations, Fig. 3 presents the difference, ΔP, between the deviation from planarity measured at ε ≠ 0% and that measured at ε = 0% for the crystalline interdigitated, crystalline side-chain disordered, and amorphous phases (X1-IDTBT and X1-PBTTT ΔP plots are provided in Fig. S4 and S5, ESI†). The interdigitated crystalline phases exhibit a symmetric response to both expansion and compression, with a maximum deviation of 2.0° for the X1-PBTTT and a standard deviation of approximately 10° (see Tables S3 and S4, ESI†). However, we observed a very distinct behavior between IDTBT and PBTTT. While PBTTT gets more (less) planar upon compression (expansion) within the lamella, IDTBT follows the opposite behavior. These distinct trends are coherent with the different torsional energy profiles exhibited by their respective dihedral angles. While the QM thiophene–thiophene torsion energy profile in the gas phase¹⁷ suggests a more twisted conformation, IDT-BT dihedral profile suggests that this polymer should be more planar. This trend is opposite to what is observed in the MD simulations in the crystalline phase where PBTTT is more planar than IDTBT, which highlights the crucial role played by intermolecular interactions in determining the polymer backbone conformation. Upon elongation, X1-PBTTT and X2-PBTTT polymorphs tend to get less planar, while X1-IDTBT and X2-IDTBT polymorphs become more planar, matching the trend of the torsion profiles. In contrast, the X2d-IDTBT and X2d-PBTTT morphologies show a similar trend in deviation from planarity but with an increased amplitude because of the more disordered microstructure as compared to their “parent” sample X2-IDTBT and X2-PBTTT. The larger deviations observed in these phases, compared to the corresponding crystalline phases, are further confirmed by an increase in standard deviation. As expected, the highest deviation from planarity is observed in the amorphous phases, where the lack of positional order prevents the formation of π-stacking interactions between the backbones of two chains (as evidenced by radial distribution functions later in this section) and allows for a wider range of torsional arrangements. The positional order of the polymeric crystalline phases was probed by calculating the radial distribution functions, g(r), at ε = 0 and ε = +1.0%, −1.0%. The radial distribution functions of a-IDTBT and a-PBTTT phases and X1-IDTBT and X1-PBTTT are shown in Fig. 4 (see ESI† for the radial distribution functions of X2-, X2d-IDTBT and X2-, X2d-PBTTT polymorphs). For the amorphous phases we herein reported only the isotropic strain ε_ISO = (ε_x + ε_y + ε_z)/3, while for the crystalline phases three different plots are shown, where each deformation corresponds to the direction of application of strain (i.e. ε_x, ε_y, ε_z). Regarding the crystalline phases of IDTBT (see Fig. 4 and Fig. S18, ESI† for X1-IDTBT and X2-IDTBT, respectively), we found that the deformation along the x axis (corresponding to cell vector a, i.e., to the spacing between the lamellar planes), mostly perturbed the position of the peaks above 25 Å, which correspond to the inter-lamellae distance in agreement with the extent of the alkyl lateral chains (∼20 Å). On the contrary, deformations along y and z (approximately corresponding to lattice vectors b and c, respectively) induced a remarkable structural modification with strain, more precisely linked to the enhancement of the intra-lamellar and the intra-chain monomer repeating unit distances in the range 5–10 Å and 12–17 Å. Among the two investigated conformers, X1-IDTBT is characterized by the largest reorganizations, which makes it less resilient to mechanical constraints due to the tighter packing of the alkyl side chains compared to that of X2-IDTBT, resulting in a greater propensity for structural reorganization. Concerning PBTTT crystalline phases (see Fig. 4 and Fig. S24, ESI† for X1-IDTBT and X2-IDTBT, respectively), we determined comparable changes between the two crystalline polymorphs. The deformation along y (roughly corresponding to b, i.e., to the distance among polymeric chains in between the lamellar plane), again alters only the characteristic inter-lamellar peaks above 20 Å, with very small contributions below 20 Å due to the non-perfect correspondence between b and y. In X2d-IDTBT and X2d-PBTTT (see Fig. S17 and S23, ESI†), the more disordered structure hampers an anisotropic dependency on the application of differently oriented strains, and furthermore, only minor changes at very short distances (first layers within a lamella) are detected. Besides, we infer X2d-IDTBT to be more flexible compared to the corresponding crystalline one (cf. E_ISO of Table 2), as it is likely associated with more free volume which allows for smooth reorganization of the polymer chains without too much perturbing their intermolecular organization. Regarding X2d-PBTTT, we detected a stronger overlap of the monomeric as a function of the applied strain, as aforesaid by commenting on the elastic properties in the previous section. For the amorphous phase, no significant differences can be observed between the expansion and compression deformations, as shown in Fig. 4.

Fig. 3 Average ΔP, i.e., the difference between the backbone deviation from planarity for ε = −1.0%, −0.5%, +0.5%, +1.0%, and ε = 0% at increasing phase ordering (from lower to upper panels: amorphous, disordered side-chain crystalline and interdigitated crystalline phases of IDTBT and PBTTT) as a function of uniaxial strains along x, y and z. The averaged deviations from planarity have been estimated from the analysis of torsion angles converted in the 0°–90° range over a 10 ns NVT trajectory.

Fig. 4 Radial distribution functions, g(r), computed between the sulfur atoms for the amorphous phases (a-IDTBT and a-PBTTT) and the interdigitated crystalline phases (X1-IDTBT and X1-PBTTT) for ε = 0%, ε = −1% and ε = +1%. Variations upon stress application are magnified 10 times (red and blue lines for ε = −1% and ε = +1%, respectively). Isotropic strain only (ε_ISO = (ε_x + ε_y + ε_z)/3) is plotted for the amorphous phases, while for the crystalline phases ε_x, ε_y and ε_z are shown separately.

Experimental nanomechanics

Experimental values for the Young's modulus, on the nanoscale, were obtained through measurements using an atomic force microscope (AFM). We used the pinpoint mode on a Park Systems NX10 atomic force microscope to do such measurements on thin films, a few tens of nanometers in thickness. Details of how such measurements are performed can be found in an earlier publication.³⁰

In the case of C16-IDTBT, it was shown through both high-resolution transmission electron microscopy (TEM) and through higher eigen mode (HEM) imaging, that its thin films contain ordered domains.^30,55 These domains are typically on the scale of a few tens of nanometers, in which the polymer backbones line up parallel to each other. It was also shown that side-chain interdigitation was absent within these ordered domains, since the gap between the polymer chains was far less compared to that expected if side-chain interdigitation was present. The tiny ordered domains in the films of C16-IDTBT represent liquid crystalline-like domains,^30,55 with a structure similar to that of the polymorph X2d-IDTBT for which calculations were performed in this work. Within the same films of C16-IDTBT, tiny domains of disorder are also present. In these domains, the backbones of adjacent polymers are not parallel to each other and no side chain interdigitation is present. These disordered domains are again on the scale of a few tens of nanometers, and come close to the amorphous a-IDTBT calculated in this work.

The regions of order and disorder are juxtaposed and scattered throughout the film, for which reason it is difficult to isolate and measure only the modulus of the ordered phase within C16-IDTBT films in the experiment. Over any scanned area, a combined contribution of ordered and disordered phases will be seen. We report the average value of the modulus for C16-IDTBT here, obtained by averaging over ordered and disordered domains in an area of a square micron.

In the case of C14-PBTTT, we measure its modulus when its film is set in a semicrystalline phase, by annealing it to 210 °C during thin film fabrication. C14-PBTTT is known to show three phases; an as-cast phase with low order due to incomplete side-chain interdigitation within small grains, a terraced semicrystalline phase with larger domains and enhanced ordering when annealed between 180 and about 210 °C, and a nanoribbon phase when annealed beyond 230 °C.^56,57 From the viewpoint of calculations, the interdigitated polymorphs X1-PBTTT and X2-PBTTT come closest to what is experimentally observed in the semicrystalline terraced phase of C14-PBTTT. For this reason, we focus on the semicrystalline terraced phase for measuring the Young's modulus.

Fig. 5 shows experimental measurements of the Young's modulus of C16-IDTBT and C14-PBTTT measured on our atomic force microscope. Fig. 5a are topographical maps of the films. These images show that on the nanoscale, the domains in C16-IDTBT are far smaller compared to the ordered terraces seen in the case of C14-PBTTT films. Fig. 5b are the corresponding Young's modulus measured over the same areas shown in Fig. 5a. The speckled blue and red image of the modulus map for C16-IDTBT demonstrates the nanomechanical texture mentioned earlier, i.e., that tiny ordered and disordered domains, with stiff and soft regions, coexist on the nanoscale. The mean values of the Young's modulus of the two polymers are shown in Fig. 5c using violin plots. These violin plots demonstrate that there are regions in the C14-PBTTT film that have Young's moduli over 4 GPa even though the mean value is just over 2 GPa. The larger values of the Young's modulus typically arise from the grain boundaries in the film for reasons mentioned in an earlier publication.³¹Fig. 5d shows a comparison between the height profile and accompanying Young's modulus profile of C16-IDTBT and C14-PBTTT extracted along the white dotted lines in Fig. 5a and b. It is evident that even though there may be particles with significant height on the nanoscale in C16-IDTBT, its Young's modulus is uniform with the regions around it. It is also clear from both Fig. 5c and d that the average values of the Young's modulus of C14-PBTTT in its ordered phase are larger than those of C16-IDTBT. This finding correlates well with the trend given by values shown in Table 2 and is in agreement with the isotropic modulus of X1-PBTTT and X2-PBTTT being larger than X2d-IDTBT. Compared to the calculated values of the Young's moduli, the discrepancy in the absolute values of the Young's moduli can be attributed to a degree of isotropic character of the thin films, which are inevitably far from the single crystal calculated ones. Furthermore, a comparison between the computational results with experimental techniques measuring the mechanical properties at the macroscopic scale, such as using buckling metrology⁵¹ and film-on-water²¹ measurements, shows broad agreement. This suggests that tensile strength measurements provide rather heterogeneous results depending on the morphology of the film (e.g., grain dimensions depending on the different manufacturing processes), in line with the more local measurements carried out with the AFM.

Fig. 5 Measured nanomechanical properties of C16-IDTBT and C14-PBTTT using an atomic force microscope. (a) Surface topography. (b) Young's modulus on the nanoscale (c) statistical distributions of the Young's modulus. (d) Comparisons of line profiles of both topography and modulus in the two polymers. Although the topography profile of IDTBT shows roughness, its corresponding modulus is uniform.

Conclusions

Exploiting the theory of linear elasticity, we developed an effective protocol for the determination of the elastic response of bulk PBTTT and IDTBT semiconducting polymers, which are widely employed in the construction of stretchable devices. We compared the mechanical response of the amorphous, crystalline interdigitated, and poorly interdigitated phases, whose investigation is rarely considered in comparison to their well-studied charge transport and optical properties. It was recognized that PBTTT is a stiffer polymer compared to IDTBT, also in the amorphous phase, with an estimated isotropic Young's modulus of about 2 GPa and about 1.5 GPa, respectively. For the crystalline structures, the crystalline packing was correlated with the anisotropic Young moduli E_x, E_y and E_z. In both polymers, the highest Young modulus was found to be the one corresponding to the backbone direction of propagation. E_x and E_y were alternatively higher according to the direction of the π-stacking interaction, which is along a for IDTBT and along b for PBTTT. Additionally, a non-negligible decrease in the elastic properties was attributed to the lack of interdigitation of the aliphatic side chains. From a combination of local and orientational order observables, we gained insights into the weak structural changes occurring after the application of a uniaxial strain, especially in the amorphous phases. Finally, we emphasize the good agreement between the experimentally and computationally determined Young's modulus values for the two polymers, highlighting that the mechanical properties originate predominantly from disordered regions in both polymers while the contribution of the most ordered regions of the film is expected to be higher in PBTTT than in IDTBT.

Author contributions

SC and DB performed the numerical calculations of mechanical properties under the supervision of YO, LM, and SO. VL provided the force fields employed in the computer simulations. KHH performed the nanomechanical measurements on a Park Systems NX10 atomic force microscope in the laboratory of DV and LF. SC, LM, YO, DV, and SO designed this theory-led, experiment-augmented project and wrote the paper jointly with KHH, and LF. All authors contributed equally to the revision of this paper.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting this article have been included as part of the ESI.†

Acknowledgements

This research used resources from the “Plateforme Technologique de Calcul Intensif (PTCI)” (https://www.ptci.unamur.be) located at the University of Namur, Belgium, which is supported by the FNRS-FRFC, the Walloon Region, and the University of Namur (conventions no. 2.5020.11, GEQ U.G006.15, 1610468, RW/GEQ2016 et U.G011.22). The PTCI is a member of the “Consortium des Equipments de Calcul Intensif (CECI)” (https://www.ceci-hpc.be). The present research benefited from the computational resources made available on Lucia, the Tier-1 supercomputer of the Walloon Region, infrastructure funded by the Walloon Region under the grant agreement no. 1910247. We acknowledge the CINECA award under the ISCRA initiative, for the availability of high-performance computing resources and support. D. B. acknowledges a grant from the “Fonds pour la formation à la Recherche dans l’Industrie et dans l’Agriculture” (FRIA) of the FRS-FNRS. D. Venkateshvaran acknowledges the Royal Society for funding in the form of a Royal Society University Research Fellowship (Royal Society Reference No. URF/R1/201590). D. Venkateshvaran also acknowledges the Wiener-Anspach Foundation for support through an FWA grant.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5tc01620g

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