Effects of network structure on the mechanical and thermal responses of liquid crystal elastomers

Two liquid crystal mesogens, 1,4-Bis-[4-(3-acryloyloxypropyloxy)benzoyloxy]-2-methylbenzene (RM257) and 1,4-Bis-[4-(6-acryloyloxyhexyloxy)benzoyloxy]-2-methylbenzene (RM82), were purchased from Wilshire Technologies Inc. (Princeton, NJ, USA). 3,6-Dioxa-1,8-octane-dithiol (EDDT), trimethylolpropane tris(3-mercaptopropionate) (TMPMP), pentaerythritol tetrakis(3-mercaptopropionate) (PETMP), 2-Hydroxy-4'-(2-hydroxyethoxy)-2-methylpropiophenone (HHMP), dipropylamine (DPA), and solvents toluene and dichloromethane (DCM) were purchased from Millipore Sigma (St Louis, MO, USA). All reagents were used in their as-received condition.

Isotropic polydomain nematic elastomers were prepared following a previously developed two-stage reaction which employs thiol-Michael addition followed by an acrylate photopolymerization [10]. An acrylate to thiol ratio of 1.15:1 was maintained for all formulations. Briefly, mesogen RM257 or RM82 was first dissolved in toluene or DCM respectively, and then heated to 80 °C (for RM257/toluene) and 60 °C (for RM82/DCM) to facilitate dissolution. Thiol spacer EDDT (85 mol%) and crosslinkers PETMP or TMPMP (15 mol%) were then added to the solution and thoroughly mixed. After cooling to room temperature, photoinitiator HHMP and a dilute catalyst solution (1:50 DPA in toluene) were added and thoroughly mixed. The mixtures were then poured into molds and reacted at room temperature for 48 h. After solidification, the solvent was evaporated at 80 °C under vacuum for 48 h.

Polydomain samples were polymerized in a Spectrolight UV light box using broadband UVA light for 10 min at 0.1 mJ cm⁻² intensity. Aligned samples were cut into 5 mm × 15 mm × 2 mm rectangles. The samples were stretched uniaxially to 75% of their tested failure strain for each formulation, followed by UV exposure, as stated above.

Polydomain samples of each formulation (n = 4) were cut into 2 mm × 20 mm × 0.2 mm rectangles. Uniaxial tensile tests were performed on an Instron 5900 with a 500 N load cell using a crosshead speed of 0.833 mm s⁻¹. After the test, peak stress and elongation at break were recorded. Elastic modulus was calculated from the linear slope of the first 2% strain of a stress–strain curve to probe the mechanical properties before the liquid crystal alignment occurred.

A differential scanning calorimeter (Mettler-Toledo) (DSC) with the FRS5 sensor under nitrogen purge at 50 ml min⁻¹ was used to determine the glass transition temperature (T_g), and nematic to isotropic transition temperature (T_ni). Prior to the experiment, the instrument was calibrated for melting temperature and enthalpy using indium and zinc metal standards. Polydomain LCE samples were cut into specimens with mass of approximately 14 mg and heated from −70 °C to 200 °C at a rate of 20 °C min⁻¹, followed by cooling to −70 °C at 50 °C min⁻¹ for two cycles. All test data were analyzed with STAR^e SW 10.0 software. T_g was defined as the midpoint of the heat flow before and after the glass transition according to ASTM D7426, and the peak temperature of the endothermic transition was used as T_ni.

LCE work capacity of aligned samples was determined using single stroke vertical weight lifting tests. The dimensions and weight of each sample were recorded before hanging the sample from a ring stand using a binder clip. A 100 g weight was hung from the bottom of the sample using another binder clip. Using a convection-based heat gun, the samples were heated to a temperature 30 °C above T_ni, ensuring full actuation. During the test, the displacement and response time of the vertical motion were recorded using a long exposure camera (Brinno TimeLapse Camera TLC200 f1.2) with a sampling rate of 1 Hz. LCE work capacity was calculated using W = F × d, where W is the work capacity, F is the force applied by the weight, and d is the distance over which the LCE pulls the weight during actuation.

Theoretical estimates of monomer length were obtained using MD simulation. Based on the molecular structure of each repeat unit, the OpenBabel toolkit [28] was used to produce a rough three-dimensional conformation in vacuum, leading to partially coiled structures. Interatomic energies and forces were then calculated more accurately using the density functional tight binding (DFTB) method as implemented in DFTB+ [29, 30], using empirical parameters fitted for organic and biological molecules with extension to include sulfur (3ob-3-1) [31]. Uncoiled structures, better corresponding to theoretical Kuhn segments in a solid polymer where the mesogen-to-mesogen interaction is considered, were obtained by calculating the energy as each monomer was quasi-statically extended by constrained displacement of the terminal atoms. For each displacement, the molecular geometry was relaxed and corresponding energy calculated by minimizing atomic force components within 10⁻⁴ eV Å⁻¹ with respect to position by the conjugate gradient method. The Kuhn length was then estimated by identifying the critical length above which the molecular repeat unit was fully extended and exhibited a sharp increase in stored elastic energy.

The work capacity of LCE can be expressed using equation (1) based on the first law of thermodynamics:

$$\begin{eqnarray}&&{\rm{\Delta }}W={\rm{\Delta }}Q-{\rm{\Delta }}U=T{\rm{\Delta }}S\,\mbox{--}\,{C}_{p}{\rm{\Delta }}T,\end{eqnarray} \tag{ 1 }$$

where W is work capacity, Q is the energy input, U is the internal energy, and S, T and C_p are entropy, temperature and specific heat, respectively.

Based on equation (1), to maximize work output, the entropy change of the system must be increased, and/or the specific heat of LCE decreased. Given the macromolecular nature of LCE, change in C_p is not always significant. As a result, enhancing work capacity by maximizing the entropy change of the LCE upon transition from the ordered to the disordered state is a reasonable approach. Based on classical rubber elasticity [34], the entropy of an elastomer can be obtained using equation (2)

$$\begin{eqnarray}&&{\rm{\Delta }}S=-\displaystyle \frac{1}{2}N\kappa T{\lambda }^{2},\end{eqnarray} \tag{ 2 }$$

where λ is the stretch ratio of the network, κ is the Boltzmann's constant, T is temperature, and N is the number of chains per volume.

According to equation (2), high entropy change can be realized by increasing the LCE network stretch ratio as well as the number of chains per volume. Assuming affine deformation, the maximum LCE network stretch ratio (λ_max) can be calculated by comparing the mean end-to-end chain segment distance $\left\langle r\right\rangle$ and the average contour length between crosslinks (τ) [35] (equation (3) and figure 1)

$$\begin{eqnarray}&&{\lambda }_{{\max }}=\tau /\left\langle r\right\rangle .\end{eqnarray} \tag{ 3 }$$

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In equation (3), the fully extended chain contour length between crosslinks (τ) is the summation of the Kuhn's length (b) for all repeat units (n) and equals nb.

The mean end-to-end distance (〈r〉) for random walk model can be obtained using equation (4):

$$\begin{eqnarray}&&\left\langle r\right\rangle =b\sqrt{n}.\end{eqnarray} \tag{ 4 }$$

As a result the overall LCE network stretch ratio (λ) relates to the number of repeat units (n) via:

$$\begin{eqnarray}&&{\lambda }_{{\max }}=\sqrt{n}.\end{eqnarray} \tag{ 5 }$$

From this analysis, it is clear that the average contour chain length between crosslinks as well as the number of chains that can be activated during stretching greatly influence the entropy change of LCE network and thus the LCE work capacity.

To explore these theoretical relationships, three experimental variables were controlled in the thiol-acrylate LCE network. An excess of diacrylate liquid crystalline mesogen was reacted with a dithiol chain extender and either a trifunctional or tetrafunctional thiol crosslinker, as presented in figure 2. The number of repeat units or the degree of polymerization (n) is varied by changing both the crosslinker content and the crosslinker functionality. The molecular weight of the Kuhn segment$\left({M}_{{w}_{b}}\right)$ was varied by changing the length of the diacrylate mesogen spacer. The two mesogenic monomers had identical rigid core components but contained different alkane chains flanking the cores. With this synthetic strategy, we can isolate the contribution of each experimental variable by constructing comparative networks. The network which will act as the basis for all comparisons consisted 15 mol% excess RM257 reacted with a thiol mixture of 85/15 ratio chain extender to a tetrafunctional crosslinker. Details of the chemistry can be found in the supplemental information table S1 is available online at stacks.iop.org/MFM/3/015002/mmedia.

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To quantitatively compare entropy of different networks, uniaxial tensile tests were performed to extract information about the experimental network in relation to the theoretical network values. First, we determined the theoretical values for ${\lambda }_{{{\max }}_{t}}$ using equation (3). Then we computed the theoretical molecular weight between crosslinks $\left({M}_{{c}_{t}}\right)$ using equation (6):

$$\begin{eqnarray}&&{M}_{{c}_{t}}=\displaystyle \frac{1}{{f}_{C}}\left[{R}_{AC}{M}_{{w}_{A}}+{R}_{BC}{M}_{{w}_{B}}+{M}_{{w}_{C}}\right],\end{eqnarray} \tag{ 6 }$$

where the subscripts A, B, and C refer to the monomeric components of diacrylate mesogen, chain extender, and crosslinker, respectively. f_C is the functionality of the crosslinker, R_XY is the molar ratio of two monomeric components, and ${M}_{{w}_{X}}$ is the molecular weight of the monomeric component. This estimation of molecular weight between crosslinks is based on equal stoichiometric ratios between acrylate and thiol functional groups and retains Flory's ideal network assumptions [36]. The number of theoretical repeat units in the contour segment between crosslinks (n_t) was estimated by comparing ${M}_{{c}_{t}}$ to the molecular weight of each Kuhn's unit $\left({M}_{{w}_{b}}\right)$. The results are shown in table 1. Table 1 also shows the Kuhn lengths (b) of both RM257+EDDT and RM82+EDDT which were estimated based on DFTB calculations, which are plotted in figure S1.

Table 1.  Theoretical network values for LCE networks.

Mesogen	Crosslinker content (mol%)	fC	${M}_{{w}_{b}}$ (g mol−1)	b(nm)	${M}_{{c}_{t}}$ (g mol−1)	nt	τt (nm)	${\left\langle r\right\rangle }_{t}$ (nm)	${\lambda }_{{{\max }}_{t}}$
RM257	4.7	4	771	4.4	1873	2.4	10.7	6.9	1.6
RM257	7.0	4	771	4.4	1312	1.7	7.5	5.7	1.3
RM257	9.3	4	771	4.4	1034	1.3	5.9	5.1	1.2
RM257	23.3	4	771	4.4	565	0.7	3.2	3.8	0.9
RM257	7.0	3	771	4.4	1719	2.2	9.8	6.6	1.5
RM82	7.0	4	855	5.4	1473	1.7	9.3	7.1	1.3
RM82	7.0	3	855	5.4	1934	2.3	12.2	8.1	1.5

The experimental maximum stretch ratio${\lambda }_{{{\max }}_{e}}$ and average molecular weight between crosslinks $\left({M}_{{c}_{e}}\right)$ values were obtained by performing uniaxial tensile tests. ${M}_{{c}_{e}}$ values were obtained using the classical rubber theory [34, 37]:

$$\begin{eqnarray}&&E\,=3\rho RT{M}_{{c}_{e}},\end{eqnarray} \tag{ 7 }$$

where E is the elastic modulus of the LCE, T is temperature, R is the gas constant, and ρ is the specific density. The molecular weight and Kuhn length were assumed the same as the theoretical values. The experimental degree of crosslinking (n_e) was calculated using the same method as described in the theoretical section above for table 1.

Comparing the theoretical and experimental values here gives great insight to the generalization of an LCE as an ideal rubber. Within both the theoretical and the experimental values, the expected trends hold. As the crosslinker content was increased, both M_c and λ_max decreased. As the crosslinker functionality was increased, both M_c and λ_max decreased. As the mesogen spacer length was increased, both M_c and λ_max increased. These qualitative relationships hold true to equation (1) and our central hypothesis that increasing chain length increases the system entropy.

Figure 3 shows the comparison between experimental and theoretical values for M_c and λ_max. The results indicate that the average molecular weight between crosslinks shows good agreement between theory predicted and experimentally measured values (figure 3(A)). However, the theoretical ultimate extension ratios significantly underestimate the experimental values due to the deviation of these LCE systems from the ideal rubber assumption.

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In addition to the mechanical performance, table 2 also shows the effects of network properties on the two thermal transitions in LCE: glass transition and nematic to isotropic transition. The glass transition is a second order transition associated with the relaxation of the chains in the amorphous phase, and the magnitude of the glass transition temperature (T_g) reflects the level of constraints that the chains are experiencing. Higher LCE network crosslinking density restricts the segmental relaxation in LCEs leading to higher T_g. The experimental results from changing the crosslinker content confirm the theoretical predictions that higher crosslinker content or higher crosslinking density enhanced the magnitude of T_g. This finding is in agreement with previous experimental results [20]. Additionally, higher crosslinker functionality increased the magnitude of T_g. Based on chain length, the RM82 network with tri-functional crosslinker should yield the lowest T_g and thus highest entropy. Experimentally, we find that this network is evenly matched with the RM257 network with the tri-functional crosslinker, illustrating that mesogen spacer length has a small effect on at least the thermal transitions of the network, and possibly the chain entropy as well. This conclusion is consistent with previously reported results, which indicate that increasing chain extender length does not affect the T_g [19]. Lastly, the nematic to isotropic transition temperature (T_ni) also decreased with increasing crosslinking density. This is likely due to the domain size decrease induced by high crosslinking density, which in turn reduced the extended crystal thickness in the nematic phase. The thinner nematic phase crystal thickness in turn leads to a lower T_ni according to the Gibbs–Thomson effect [38].

Table 2.  Experimental data for LCE network structures.

Mesogen	Crosslinker content (mol%)	fC	${M}_{{w}_{b}}$ (g mol−1)	${\lambda }_{{{\max }}_{e}}$	${M}_{{c}_{e}}$ (g mol−1)	ne	Tg (°C)	Tni (°C)
RM257	4.7	4	771	3.2	1412	1.8	3.7	70.1
RM257	7.0	4	771	2.6	913	1.2	5.8	71.8
RM257	9.3	4	771	1.9	784	1.0	7.5	68.4
RM257	23.3	4	771	1.9	135	0.2	18.2	66.0
RM257	7.0	3	771	3.1	1782	2.3	0.7	65.9
RM82	7.0	4	855	3.9	1177	1.4	−9.2	91.7
RM82	7.0	3	855	4.4	1420	1.7	−10.3	86.0

Here, we return to the central hypothesis of the paper: the network work capacity can be increased by increasing the chain entropy. While qualitative and quantitative manipulation of chain entropy was demonstrated in the preceding sections, the effect on work capacity has not yet been proven. Based on equation (1), changing the entropy should change the amount of work done by a phase change. As discussed in section 3.1, work capacity should vary with entropy as a function of two main variables: the ultimate extension (λ_max) and the molecular weight between crosslinks (M_c). Work capacity of each LCE network is determined by a weight lifting test detailed in the methods. As the weight attached increases, the work capacity increases as well. Figure 4 compares the work capacity of different LCE networks as a function of both theoretical and experimental ultimate extension, or molecular weight between crosslinks. Work capacity is measured for two loading stresses: σ = 1 MPa and σ = 2 MPa. From equations (1)–(3), (5), and assuming a constant C_p, we can expect the following relationship:

$$\begin{eqnarray}&&{\rm{\Delta }}W\propto \displaystyle \frac{1}{2}N\kappa {T}^{2}{{\lambda }_{{\max }}}^{2}-{C}_{p}{\rm{\Delta }}T.\end{eqnarray} \tag{ 8 }$$

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Therefore, λ_max should have a power law relationship with work capacity. Indeed figures 4(A), (B) show a power relationship for the theoretical extension ratio. Figures 4(C), (D) show that the increase in M_c generally increased the work capacity of LCE even though the theoretical data seemed to show maxima at M_c around 1500 g mol⁻¹. One possible explanation is the quality of mesogen alignment, which is not accounted for in classical rubber elasticity, as it is uniquely liquid crystalline. It is well known that the quality of alignment affects the amount of contraction produced by the LCE [39, 40]. Therefore the quality of the liquid crystalline alignment is a large enough factor to influence the patterns of small sample sets. Regardless of this liquid crystalline contribution, classical rubber elasticity did predict the correct general trends for the LCE work capacity.

Therefore, using classical rubber elasticity is a valid method to guide the design of high work capacity LCEs, and can perhaps be applied to other traits, such as power. However, the effects of liquid crystalline behavior cannot be neglected if specific values are targeted for work capacity, as the liquid crystalline contributions to the entropy are too high to discount. It should be noted that the theoretical maximum work capacity demonstrated in figures 4(a) and (c) of approximately 150 kJ m⁻³ should not be taken as a benchmark maximum work capacity value. Though the maximum for the variables tested in this work, there are many factors which contribute to the maximization of work capacity not covered in this work. These factors include applied stress, chain orientation, energy conversion efficiency, and breaking strength, among others [17, 41–44]. However, the central finding of this paper should remain consistent over all networks, that maximizing the entropy though network structure can maximize the work capacity of a given system.