1. Introduction

Efficiency of the second harmonic generation (SHG) in poled thin films of functionalized polymers, consisting of quasi 1D charge transfer (CT) chromophores in a polymer matrix depends on degree of polar order of CT molecules, induced by electric field corona poling, photo-assisted electric field poling or all optical poling methods [1]. The second order nonlinear optical (NLO) susceptibility is characterized in this case by two tensor components (for details see e.g., Ref. [2]): the diagonal

and the off diagonal one

where Z is the poling field direction (perpendicular to the thin film surface) and X is perpendicular to it. θ is angle which makes the molecular axis z with Z direction. N denotes the number density of the chromophores (dipoles), β_zzz – molecular first hyperpolarizability of CT chromophores, F – the local field factor. In Eqs. (1) and (2) brackets 〈…〉 denote configurational average over all chromophore orientations.

Load parameter defined as 𝓛 = N < cos³ θ >, which is proportional to χ ⁽²⁾ _ZZZ, plays the central role in our study. Formula (1) predicts a linear dependence of χ ⁽²⁾ _ZZZ on N. Departure from linear law is known since the last decade [3, 4]. Linear behaviour is reported at small concentration only, say, for N < 10²⁰/cc. For higher concentrations saturation and, occasionally, subsequent falloff of SHG susceptibility are observed in guest–host [5, 6, 7, 8] chromophore – polymer systems. Those effects are caused by a few competing mechanisms which become more or less pronounced depending on the system [9]. On microscopic/mesoscopic scales the diminution of SHG susceptibility is due to spatial and orientational local and global correlations of chromophores, ascribed to hypothetical aggregation of dipoles into antiparallel pairs, which diminishes acentric order parameter < cos³ θ > (see Refs. [10, 11] and references cited therein).

Theoretical statistical mechanics and MC studies were originated by Dalton and co–workers [12, 13]. Subsequent lattice [14] and off–lattice [15, 16] MC simulations, Molecular Dynamics simulations [17, 18], density functional theories [19], mean field theories [20, 21], fully atomistic modeling [22, 23, 24], extended dipole models [25], or inclusion of matrix into MC simulations [26] have substantially deepened our understanding of the underlying physical processes, summarized in terms of various paradigms [10, 11]. On the other hand, they have led to contradictory results concerning aggregation hypothesis and existence/nonexistence of SHG susceptibility maximum on concentration curve [11].

Recently, optical polarization microscopy studies have revealed the importance of temporal aspects of aggregation–driven poling [9], emphasizing the role of so far completely disregarded aspect of poling – the pre–poling history of the sample.

The aim of this paper is to elucidate the role of pre–poling history in build–up of polar phase and SHG signal strength for electric field poled system of dipoles in a polymer matrix, using MC modeling. The importance of pre–poling history of the sample was first formulated quantitatively, to the best of our knowledge, in Ref. [11].

2. Maximum of SHG susceptibility: experimental studies

SHG signal was measured from electric–field poled polymethyl methacrylate - Disperse Red 1 (PMMA-DR1) side–chain polymer system. Side chain polymers are superior compared to guest–host systems in terms of stability and film quality [27, 28]. Moreover, the use of side-chain polymers secures more reliable experimental data, as the relaxation of the polar order is much faster in guest–host system because of a faster chromophore diffusion.

The polymers syntheses and characterization are described in Ref. [29]. The studied polymer thin films were prepared by spin coating of well filtered polymer solutions in 1,1,2 tetrachloroethane on carefully cleaned glass substrates. This solvent is reputed for giving good spun films. The typical polymer solution concentration was of about 120 g/l. After the deposition thin films were dried in an oven at 80°C for 1 hour. The thin films thicknesses, after the solvent removal, were determined using a commercial profilometer with a nominal precision of 1 nm. The films were heated to the glass transition temperature (T_g = 110°C) and then the poling field was applied [30]. After the poling the film was cooled down to the room temperature under the applied electric field. For all studied films the same poling conditions were applied, i. e. the distance of needle from the thin film surface of 15 mm, HV of 6 kV, the poling time of 3 minutes. It was sufficient to get the maximum poling. Our previous studies have shown that these conditions are optimal for these polymers. The optical quality of thin films is decreasing with the chromophore concentration and is particularly well seen with an optical microscope for concentrations starting from 50 mol %.

Immediately after the poling the films were mounted on a goniometer for NLO characterization by the optical SHG technique [2], performed at 1064.2 nm fundamental wavelength with a Q switched Nd:YAG laser. The data were calibrated with SHG measurements on an y-cut α-quartz single crystal, performed at the same condition, and screened using the formalism described in [2], which takes account of the harmonic wave reabsorption. More details can be found in [29].

The optical absorption of thin films was measured with a Perkins-Elmer UV-VIS-IR spectrometer. The measured absorption spectra for different concentrations are compared in Fig. 1. The optical densities of the thin films were not normalized for film thickness and thus do not scale linearly with concentration. A small blue shift of the maximum absorption wavelength λ_max is observed. However it is significantly smaller than observed for the same molecule in other matrices [31, 32]. But usually the polarity of molecule environment plays an important role in the position of excited levels, not only the aggregation. The values of λ_max for different concentrations are listed in Table 1.

The measured values of χ ⁽²⁾ _ZZZ/2 and χ ⁽²⁾ _XXZ/2 tensor components for the studied thin films as function of the chromophore molar content expressed in percents, are shown in Fig. 2 for both tensor components, expressed in the commonly used notation: d_sp = χ ⁽²⁾ _XXZ/2 and d_pp = χ ⁽²⁾ _ZZZ/2. In both cases the maximum of SHG susceptibility is present.

 

Fig. 1. Absorption spectra of thin films of PMMA–DR1 with different chromophore concentrations. The optical densities of the thin films were not normalized for film thickness and thus do not scale linearly with concentration.

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Table 1. Maximum absorption wavelength for studied thin films for different DR1 concentrations.

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Fig. 2. Molar chromophore concentration (in percents) dependence of d_sp = χ ⁽²⁾ _XXZ/2 and d_pp = χ ⁽²⁾ _ZZZ/2 for the studied thin films. The error bar (not shown) is 10%.

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In nonlinear optics and in the aggregation process important is the density of dipole moments which corresponds to the density of active molecules (cf. Eqs. (1) – (2)). This is why we preferred to use description in term of molar concentration. This follows directly from the usually used chemical formula for such polymers which in present case reads: MMA_1−xDR1_x, where MMA is monomer unit and x, multiplied by 100, gives directly the mol concentration in percents. The transformation to weight concentration is given by the following equation:

where N_seg and N_mon denote the number of substituents with dye polymer repetition units and the number of monomers, respectively. M_dye, M_seg and M_mon are, respectively, molecular masses of dye, substituted segment and monomer molecules, respectively. In the present case M_seg = 382.42 g/mol, M_mon = 100.12 g/mol and M_dye = 313.33 g/mol (314.34-1.008 because of substitution). The molar concentration refers also directly to the density of interacting dipoles, as used in MC stimulation.

Within the experimental accuracy we observe an almost linear increase of the both d_sp and d_pp tensor components up to 15 mol % of DR1 and a decrease at higher concentrations. For concentrations starting from 35 mol % both tensor components stabilize and remain almost constant. Although there are only two measurements points on the rising part of experimental data (8 mol % and 15 mol %), de facto there is another one corresponding to the origin (zeroth concentration) at which the SHG susceptibility is equal to zero. We note here that both tensor components exhibit similar concentration dependence. Also the ratio d_pp/d_sp ≈ 3 remains almost constant within the experimental accuracy which is roughly of ±20% for each concentration (the 20% relative error originates from adding 10% error on d_sp and d_pp determinations). It doesn’t take account of a possible effect of error in d_sp determination on the deduced value of d_pp. In fact the d_pp value depends on the value of d_sp injected in calculations (for details see Refs. [2, 33]).

The same polymer system was studied also by Robinson and Dalton [13] and the authors report a continuous increase of SHG susceptibility up to the concentration of 30 wt %, without reaching the maximum, with a tendency to saturate. Because of large molecular mass of DR1 molecules as compared to that of monomer it corresponds to a small substitution level of about 13.2 mol %. So their data lie on the increasing part of our experimental concentration dependence. Intriguing is a fast decrease of χ ⁽²⁾ susceptibility and its stabilization for concentrations starting from 35 mol %. A similar behaviour is observed in the case of solution of lyotropic liquid crystals (LC) in water [34]. At low concentration only monomers are present. Their concentration increases linearly with LC concentration. At higher concentration micelles (a kind of aggregates) appear and their concentration is increasing rapidly, while the monomer concentration stabilizes and remains constant when increasing LC concentration. It means that all introduced LC molecules either attach to micelles or form new ones. The mechanism behind this behaviour, similarly as in the present case, is the strong dipole-dipole interaction.

3. Monte Carlo study of guest–host system

We have simulated an off-lattice dipolar system of point dipoles with soft shell repulsion immersed in a polymer matrix, close to the glass temperature, using standard MC simulations. Polymer matrix, modelled using bond–fluctuation method, consisted of 24000 polymer chains, each of 20 monomers. We use the same specific polymer length as in our previous studies on poling and inscription of diffraction gratings [11].

The potential energy of the system consists of three parts: point dipole–dipole interaction, interaction of dipoles with poling field and repulsive soft–sphere interaction:

where E⃗ denotes electric poling field, µ⃗_i – i-th dipole moment, r_ij and r̂_ij – respectively distance and unit vector between dipoles i and j, ε – dielectric constant of the host, and σ – characteristic scale for soft–sphere interactions. The parameters were µ = 26 D, σ = 7 Å, ε_LJ/(k_BT) = 0.1, ε = 4, E = 150 V/µm, T = 350 K. Cutoff for soft core and electrostatic interactions was 5 nm. The reaction field method was also used; both approaches yield similar results.

This specific set of parameters was taken after a seminal paper on poling [15]. The dipole moment is larger than the dipole moment of individual DR1 molecule (≈ 7.5 D). It is well–known [13] that in lattice models sufficiently large values of µ are necessary to get the maximum on the susceptibility curve. We note that PMMA is a polar medium. Its microscopic effective dielectric constant is of 22–30 as reported in Ref. [35]. Functionalizing PMMA with DR1 molecules will create a more polar environment because of large dipole moment of DR1. For this reason the dipole moment of DR1 molecule in PMMA-DR1 system may be larger (“dressed” dipole) than reported in literature for individual molecule. The observed blue shift (Fig. 1) may be due to the increasing polarity of the medium, as observed when poling PMMA-DR1 system [36], counterbalancing the expected red shift due to the aggregation process [9]. Specific choice of µ is not critical since our main goal is to reproduce rather qualitatively than quantitatively the effect of maximum of χ ⁽²⁾.

The dipolar component in the polymer matrix was simulated using a minor modification of the approach proposed in Refs. [11, 25, 26]. Briefly, each of the dipoles took a trial move, either orientational or translational. The spatial distribution of monomers determined the steric interactions for dipolar system. Prior to a MC trial move of a dipole, the steric effects were taken into account via a simple, binary decision rule, which either allows or forbids MC move. In papers [11, 25, 26] the probability of translational trial movements was calculated accounting for an actual environment of the dipole. On the contrary, in this study we account for the environment in the trial position of the dipole. This choice offers a much more effective sampling of configurational space of the dipoles. For this reason a direct quantitative comparison of previous and current results might be misleading.

We study the MC evolution of SHG susceptibility χ ⁽²⁾ _ZZZ ∝ 𝓛 in two cases. In the first one, referred to as the case with pre–poling history, the poling field is switched on following an interval of electric field–free evolution during which the pre–poling processes take place. In the second case, without pre–poling history, the poling field is switched on immediately after preparing of the sample. The dependence of the amplitude E of the poling field on number of MC Steps (MCS) is shown in Fig. 3. In this case the length of the pre–poling interval 𝒫 is 10⁵ MCS.

 

Fig. 3. Poling electric field as function of number of MC steps: system with pre–poling phase (thin solid line) and without pre–poling phase (dashed line).

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4. Pre–poling history and build–up of SHG susceptibility

In Fig. 4 we show the results for the load parameter N < cos³ θ > ∝ χ ⁽²⁾ _ZZZ as a function of number density N. We concentrate mainly on higher values of N rather than on lower values which are not directly related to the maximum of SHG susceptibility. Standard errors (standard deviation of mean value) obtained from multiple independent MC simulations are of the size of graphical symbols used in the figure. The simulations support the pre–poling history paradigm. The system with pre–poling history (left) develops a maximum on 𝓛 ∝ χ ⁽²⁾ _ZZZ curve approximately at number density N_m = 1.7 × 10²⁰/cc. The poling field was switched on after elapsed 10⁵ MCS. The SHG susceptibility component χ ⁽²⁾ _ZZZ for a system without pre–poling history follows a different pattern as function of N (right). No traces of maximum are found; instead a saturation takes place again close to N_m = 1.7 × 10²⁰/cc. The size of SHG susceptibility, given by 𝓛(N_m) is approximately twice larger for the system without pre–poling history. This indicates that in the pre–poling electric field–free phase the system develops complicated dipole/polymer structures which have a strong influence on kinetics of poling and on equilibrium properties of polar phase.

 

Fig. 4. Plot of load parameter N < cos³ θ > ∝ χ ⁽²⁾ _ZZZ as a function of number density N: with pre–poling history (left) and without pre–poling history (right) (note different scales on vertical axes).

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Fig. 5. MC kinetics of acentric order parameter < cos³ θ > for N = 2.16 × 10²⁰/cc. System with pre–poling history (left) and without pre–poling history (right). Solid lines: stretched–exponential fits.

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To investigate the “microscopic” origin of two scenarios we have studied the system for N = 2.16 × 10²⁰/cc which is slightly above N_m. Fig. 5 shows the evolution (MC kinetics) for acentric order parameter A = < cos³ θ >. For the system with pre–poling history (left) small fluctuations around an overall isotropic distribution of orientations of dipoles occur in the pre–poling period. After turning the poling field on parameter A increases gradually following an active growth in the first 2 × 10⁵ MCS; the saturation value is A_s ≈ 0.15. The pattern of evolution of A in the case of direct poling (right) is very different. The initial quick evolution of A stops rather abruptly after 4 × 10⁴ MCS. In the remaining period some traces of a very weak increase are observed. The saturation value is approximately three times higher than in the previous case, A_s ≈ 0.45. The kinetics of build–up of polar phase in the system with pre–poling history follows the stretched–exponential law < cos³ θ > (t) ∝ exp(−(t/τ)^α) with α ≃ 0.5. Stretched–exponential fit in the case of immediate poling fails to reproduce the data in the very short interval after switching electric field on.

 

Fig. 6. Configuration of dipoles: initial configuration (a); after 8 × 10⁵ MCS, without pre–poling history (b); after 10⁵ MCS without electric field (pre–poling period) (c); after 8 × 10⁵ MCS for poling started after 10⁵ MCS (d). Polymeric chains are not shown.

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Different types of kinetics are intimately related to spatial and orientational organization of dipoles. Fig. 6 shows chosen configurations of dipoles during poling. The starting configuration consists of isotropically oriented dipoles in the nodes of a fluctuating cubic lattice (case (a)). The configuration after 8 × 10⁵ MCS (case (b)) of the system without pre–poling history displays a complex system of strings mostly oriented along the direction of the poling field (z axis) with prevailing head–to–tail dipolar organization, which contribute coherently to acentric order parameter < cos³ θ >. Strings perpendicular to the field are also present. At the end of pre–poling interval (case (c)) a complex spatial organization of strings appears. On the average the system does not display polar–phase organization: < cos³ θ > ≃ 0, see Fig. 5 (left). The configuration after 8 × 10⁵ MCS, with poling field turned on after 10⁵ MCS, bears more resemblance to the starting configuration (case (c)) than to final configuration for immediate poling (case (b)). The strings which were entangled during pre–poling interval cannot easily reorient themselves along the direction of poling field and hinder the build up of polar phase and of large SHG susceptibility.

5. Discussion

 

Fig. 7. Plot of normalized susceptibility χ ⁽²⁾/χ ⁽²⁾ _max against normalized concentration of dipoles (see text). 〇: experiment, ×: MC simulations, thick solid line: after Ref. [8], thin solid line: after Ref. [6].

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Direct comparison of Fig. 2 and Fig. 4 clearly shows that experimentally observed behaviour corresponds to that of a system with pre–poling history. The measured versus predicted SHG coefficients are shown in Fig. 7, where additionally the data from earlier studies of guest–host [6, 8] systems are shown. We use normalized concentrations defined as ratio of concentration to the concentration at maximum of χ ⁽²⁾. Susceptibilities were normalized to the maximum of χ ⁽²⁾: χ ⁽²⁾ _norm = χ ⁽²⁾/χ ⁽²⁾ _max. Solid lines are fits to the experimental data. In the interval of normalized concentrations where the data are available in all four cases in Fig. 7 the curves are close to each other. This indicates some kind of universality in poling of thin films of guest–host and side–chain functionalized polymers for those concentrations. MC modeling offers a reasonable description of maximum on χ ⁽²⁾ curves. For larger values of normalized concentration (> 1.5) experimental data are lower than MC predictions. This implies that for those concentrations the model does not account for some unknown physical processes responsible for spatial and orientational ordering of the dipoles in the matrix.

The plots shown in Fig. 4 indicate that the length 𝒫 of pre–poling interval is an important kinetic parameter which determines the quality of emerging polar phase in the course of poling. To investigate this topic quantitatively we have studied the dependence of equilibrium load parameter 𝓛 on 𝒫 for two reduced densities: N = 3.43 × 10²⁰/cc and N = 5.12 × 10²⁰/cc, both larger then N_m = 1.7 × 10²⁰/cc. The results for N = 5.12 × 10²⁰/cc are shown in Fig. 8. We find a strong impact of pre–poling interval length 𝒫 on value of load parameter for short pre–poling intervals. With increasing value of 𝒫 the dependence becomes weaker and the load parameter levels off after (3 − 5) × 10⁴ MCS. Similar results were found for lower number density N = 3.43 × 10²⁰/cc. We conclude that the value 𝒫 = 10⁵ MCS used in Section 4 is sufficient to account for important rearrangement processes in the pre–poling phase.

Another interesting topic is how the kinetics of the build–up process of polar phase depends on the dipole density. For reduced density N = 2.16 × 10²⁰/cc the system with pre–poling interval length 𝒫 = 10⁵ MCS follows stretched exponential kinetics with exponent α ≃ 0.5 (Section 4). Preliminary analysis shows that the kinetics for the systems with the same 𝒫 remains stretched exponential for all reduced densities studied in this paper. More specifically, the exponent α becomes dependent on the reduced density. For N < N_m = 1.7 × 10²⁰/cc α is density independent: α ≃ 0.5, while for N > N_m a slow decrease of α is observed. For N = 5.12 × 10²⁰/cc we have found α ≃ 0.35.

 

Fig. 8. Equilibrium load parameter 𝓛 = N < cos³ θ > as function of pre–poling interval length 𝒫 for reduced density N = 5.12 × 10²⁰/cc.

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Finally, let us comment briefly on the influence polymer chain length on the susceptibilities. We expect some weak dependency when the polymer chain length increases a few times, say up to 100 monomers (Kuhn elements), in a distant analogy with a weak dependence found in the case of diffraction gratings inscription in two-dimensional chromophore – polymer matrix guest–host system [37].

6. Conclusions

SHG from electric–field poled PMMA – DR1 side–chain system displays an undesirable effect: a maximum and a subsequent fall–off of SHG susceptibility as function of number density of DR1 dipoles. This effect was studied using MC modeling. We have found that presence (or absence) of maximum is conditioned by the pre–poling history of the sample. Pre–poling effects result in (i) slow, stretched–exponential build–up of polar phase; (ii) low SHG signal; and (iii) presence of a maximum of χ ⁽²⁾ _ZZZ as function of number density of dipoles. On the contrary, the build–up of polar phase in freshly prepared films without pre–poling history (i) occurs quickly and follows a more complicated non–exponential relaxation; (ii) yields larger SHG susceptibility; and (iii) results in a saturation rather than a maximum of χ ⁽²⁾ _ZZZ. On the microscopic/mesoscopic scale the origin of different kinetics is due to the spatial entanglement of linear (and more complicated) structures of dipoles with head–to–tail order in the pre–poling interval which hinders the reorientation of the dipoles, the build–up of well–ordered polar phase, and of large SHG susceptibility; stretched–exponential kinetics was observed.

Polymers (with guest inclusions) belong to the class of complex systems [38]. Thus, it is reasonable to assume that the occurrence of maximum is driven not by a single fundamental mechanism but rather by a few different mechanisms promoting various patterns of dipolar aggregation like, e.g., parallel or antiparallel aggregates. These mechanisms become more or less pronounced depending on the details of the system [9]; in particular, the tendency of the molecules to aggregate can be reduced when intermolecular interactions between a polymer host and guest dye molecules are controlled [31]. If parallel, antiparallel and more complex types of aggregates are present on various spatial scales then UV-Vis absorption spectra may display more complex features than a blue or a red shift.

Spatial inhomogeneities of any origin (kinetic or equilibrium) on sufficiently large scale can cause similar effects. The sample storage and processing conditions can have a profound effect on bulk material SHG response. The pre–poling time interval which can strongly influence (decrease) the SHG susceptibility can be relatively short. MC kinetics (Fig. 5, left) shows that the pre–poling interval constituting approximately 10% of the poling interval results in a strong decrease of SHG susceptibility. In our experiment poling took three minutes which implies that important spatial/orientational dipolar reorganization can take place in less than one minute close to T_g. The presence of dipolar aggregates in freshly obtained thin films reported in [9] may be due to similar effects in a more dense system, where linear organization is strongly suppressed.

Let us comment on MC simulations. The experimental data are for a side–chain polymer whereas the simulations, for the sake of simplicity, are performed for the guest–host system. The comparison of available experimental data for guest–host and side–chain systems in Fig. 7 shows, however, that the normalized susceptibilities are similar, and that guest–host MC modeling is acceptable. For higher normalized concentrations it is no more true and modifications of the model are necessary, including more sophisticated MC simulations and more realistic treatment of physical dipoles. This is, however not the central point of interest of our study, which is primarily oriented onto an explanation of maximum of SHG susceptibility in terms of pre–poling history of the sample.

Finally, we have shown that the maximum of SHG susceptibility depends on the pre–poling history of the sample using some set of parameters for guest–host system, like volume fraction of dipolar material and polymer chain length. Very important is also the point–dipole approximation for physical dipoles. A systematic study, including additionally extended–dipole paradigm [10, 11] and interpretation of stretched exponential kinetics in terms of “microscopic” models [39] requires massive simulations and is at progress now; the results will be published elsewhere.