Secondary relaxations and electrical conductivity in hyperbranched polyester amides



Broadband dielectric spectroscopy is used to investigate molecular dynamics and charge transport in three hyperbranched polyester amides with hydroxyl, phenyl, and stearate terminal groups. At higher temperatures, the dielectric spectra are interpreted in terms of hopping conduction in a spatially randomly varying energy landscape, whereas two secondary dipolar relaxations attributed to librations of the terminal and amide groups dominate the low temperature regime. Despite a shift of more than 3 decades in the dc conductivity upon variation of the end groups, the Barton–Nakajima–Namikawa relation is shown to hold. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 1651–1657, 2010


Hyperbranched polymers are promising for technological and fundamental applications because of their novel, high density of surface functional groups, nanoscale size, low dispersity, high degree of branching, globular and unentangled structures. They are currently under investigation for use in anticancer drug delivery systems, vaccines, antiviral, antibacterial therapeutics, powder coatings, flame retardant coatings, and barrier coatings for flexible packaging. It has been demonstrated that the chemical nature of the end group functionalities of a hyperbranched polymer dominates not only the material's solubility in various solvents1–5 but also melt and thermal properties such as the glass transition temperature.6–10 For practical purposes, the influence of the terminal groups on diffusion as well as molecular dynamics in these polymers needs to be understood over possibly broadest range of frequencies and temperatures. Because of its ability to probe molecular fluctuations and charge transport in broad frequency and temperature ranges, Broadband Dielectric Spectroscopy (BDS) turns out to be the ideal experimental tool for probing charge transport and molecular dynamics in hyperbranched polymers.

Description of the synthesis methodologies for the preparation of a wide variety of hyperbranched and dendritic polymeric systems through condensation, addition, or ring-opening reactions can be found in a number of recent reviews.11–16 These include polyesters,17–20 polyamides,21–23 and poly(ester- amides)24–27 because of their similarity in branching. Compared to dendrimers, hyperbranched polymers are more economical due to the one-step synthesis.

BDS measures the complex dielectric function, ε*, which is equivalent to the complex conductivity function, σ*. This is expressed as σ*(ω, T) = iε0ωε*(ω, T), implying that σ′ = ε0ωε″ and σ″ = ε0ωε′ (ε0 being the vacuum permittivity and ω the radial frequency). The two forms emphasize different facets of the same process. BDS has been used in the investigation of the molecular dynamics of dendrimers28–32 and hyperbranched33–36 polymers. For many hyperbranched polymers, the dielectric spectra at higher temperatures are dominated by conductivity contribution, which masks the structural α-relaxation process. This has been demonstrated in our recent article in which we investigated charge transport and dipolar relaxations in two polyamide amines of different backbone structures (different nitrogen content).37

In this article, the dielectric properties of three hyperbranched polyester amides (PEAs) with hydroxyl, phenyl, and stearate terminal groups are studied by BDS and Differential Scanning Calorimetry (DSC). The influence of the end groups on charge transport and secondary relaxations are investigated in detail.



Polycondensation of phthalic anhydride or maleic anhydride as A2 monomers and diethanol amine as B′B2 monomer yields hydroxyl functional hyperbranched PEAs. Gelation was avoided by using higher concentrations of B′B2 monomer and short reaction times. Fully soluble resins were obtained and characterized using NMR and Fourier Transform infrared spectroscopy (FTIR). Partially, stearic acid ester and benzoic acid ester functionalized diethanol amide-based hyperbranched PEAs have been synthesized in a straightforward polycondensation reaction using calculated amounts of acid anhydride, diethanol amine, and the modifying agent. Scheme 1 shows the chemical structure of the PEA with OH as terminal groups. It was obtained through substitution of OH by stearate and phenyl groups. This was confirmed using NMR and FTIR techniques. Synthesis and modification of the investigated polymers are described in further detail elsewhere.24

Scheme 1.

Chemical structure of hyperbranched polyester amide with hydroxyl terminal groups.

Nuclear Magnetic Resonance

Aromatic Hyperbranched PEA Modified with Phenyl: PEA-Ph

Hyperbranched PEA is partially modified using benzoic acid. The 1H NMR spectrum shows that the polymer still contains linear and terminal OH end groups. It is also observed that the peaks are multiplet due to the presence of the linear and terminal isomers, and, in the aromatic region due to presence of two different aromatic rings one from the benzoic moiety and the second from the phthalic acid used to prepare the hyperbranched polymer.

1H NMR (δ d6 = ppm): 7.2–8.2 (aromatic H1:H4), 3.4 = H5, 3.15 = H6, 3.0 = H11, 3.6 = H10, 4.5 = OH, (l, t), 3.9 = H8, 3.8 = H7.

Aromatic Hyperbranched PEA Modified with Stearic Acid: PEA-Stearate

The following NMR and FTIR results were obtained:

1H NMR (DMSO-d6) δ ppm: 8.17–7.15 (aromatic ring, disubstituted), 5.58 (OH), 4.48 (CH2[BOND]OCO, l), 4.22 (CH2[BOND]OCO, d), 3.67 (CH2 OH), 3.44–2.99 (N CH2 t, l), 2.2, 2.3 (esterified stearic acid) 2.18, 1.48, 1.23 and 0.85 (stearic acid alkyl chain).

FTIR spectra: 2917 cm−1as, CH2), 2849 cm−1s, (CH2)16), 1464 [δas, CH3, δs (CH2)16 ], 1377 cm−1s CH3).

Differential Scanning Calorimetry

DSC measurements were carried out in the temperature range −80 to 80 °C using a Perkin–Elmer Pyris 6 calorimeter. Any previous thermal history of the samples were erased by performing a first heating run up to 10° above Tg and cooling down with a cooling rate of 10 °C/min. The second heating scan was then recorded.

Broadband Dielectric Spectroscopy

The dielectric measurements were carried out by means of a Novocontrol high resolution alpha dielectric analyzer in the frequency range 0.01 Hz to 10 MHz. The analyzer was supported by Quatro temperature controller using pure nitrogen as heating agent and providing a temperature stability better than 0.2 K. The measurements were conducted, using gold-plated stainless steel electrodes of 20 mm in diameter, in parallel plate capacitor configuration. The samples were annealed at temperatures above the calorimetric glass transition temperatures in vacuum for 24 h before performing dielectric measurements.


The dynamic glass transition is determined by intermolecular interactions and shows up differently in the various physical quantities such as dielectric relaxation, viscosity, and heat capacity. In polymers, it depends on molecular structure, molar mass, degree of branching, nature of the end groups, and other influences like hydrogen bonding within the polymer structure. The physical parameters can be probed to obtain detailed information concerning dynamic structural processes and charge transport in hyperbranched polymers.

DSC measurements are often used to estimate the calorimetric glass transition temperature, Tg, of the hyperbranched polymers. The glass transition temperature depends on a number of factors, which affect rotation of chain links, mobility, and chain–chain interaction. These include molecular structure, molar mass, degree of branching, nature of the end groups, and interactions like hydrogen bonding within the polymer structure. DSC using heat program from −80 up to 40 °C, with heating rate of 10 K/min, was used as shown in Figure 1. The Tg was calculated from the midpoint of the heating trace in the transition range of each sample. The results of Tg were 241, 259, 273 K ± 2K for PEA-OH, PEA-Ph, and PEA-Stearate, respectively. This remarkable change in the Tg upon variation of the terminal groups can be attributed to the differences in packing as a result of the end groups.

Figure 1.

Differential Scanning Calorimetry heating thermogram for hyperbranched polyester amides with OH, Stearate, and Phenyl terminal groups. Inset: Further heating reveals a melting point accompanied by a temperature of crystallization upon cooling in case of stearate modified polymer. The rate of heating and cooling was 10 K/min.

The hydroxyl terminated polymer has the lowest Tg. The modification of the aromatic polymer by long alkyl chains (stearate) instead of hydroxyl groups shifted the Tg to relatively higher temperature. Further extrapolation of the heat program up to 80 °C in case of (PEA-Stearate) is illustrated graphically in the inset of Figure 1. Melting temperatures, Tm, upon heating and of crystallization, Tc, in the cooling cycle are observed in the range of 40–60 °C, respectively. This reflects the influence of the long alkyl chains in creating an ordering or even semicrystalline structure in the polymer.38 The ordering of the long chains in the polymer backbone reduces the free volume in the polymer. The reduction of the free volume is affected by restricting the molecular motion because of the higher occupied volume.

Figure 2 shows the dielectric (and conductivity) spectra for PEA-OH polymer. At higher temperatures, charge transport dominates the spectra, whereas secondary dipolar relaxations are observed at lower temperatures. Pronounced changes in the complex conductivity function are detected due to electrode polarization at lower frequencies. This process has been shown to exhibit a characteristic dependence on the length of the sample cell and the material of electrode used. It is studied in further detail in ref.41.

Figure 2.

(a) The complex dielectric function (ε*(ω) = ε′(ω) − iε″(ω)) and complex conductivity function (σ′(ω) = σ′(ω) + iσ″(ω)) versus frequency at various temperatures (in the range 323–243 K) as indicated for hyperbranched polyester amide with hydroxyl end group (PEA-OH). Inset: Scaling with respect to the characteristic frequency, νc, and dc conductivity, σ0, for the different temperatures. Lines denote representative fits based on the random barrier model proposed by Dyre (eq 1), and (b) the frequency dependence of the complex dielectric function (ε* (ω) = ε′(ω) − iε″(ω)) of PEA-OH illustrating the secondary relaxations at lower temperatures. The lines are fits of the Havriliak–Negami equation (eq 2). The error bars are smaller than the size of the symbols, if not specified otherwise

Charge transport in disordered solids has been shown to be well described by a theoretical model proposed by Dyre.39, 40 Within this approach, conduction takes place by hopping of charge carriers which are subject to spatially randomly varying energy barriers. Solved within the Continuous–Time–Random–Walk approximation, the analytical expression obtained for the complex conductivity is given by

equation image(1)

where τe denotes the time corresponding to the attempt rate to overcome the largest barrier determining the dc conductivity, σ0. From recent studies,42–44 it has been demonstrated that ωe(=1/τe) ≅ ωc, where ωc is the characteristic frequency at which the real part of the conductivity begins to increase with frequency from the σ0. The fits obtained by applying eq 1 to the dielectric spectra of PEA-OH are included in Figure 2(a) and the parameters given in Table 1. Using σ0 and ωc as the main directly observable quantities characterizing charge transport, the dielectric spectra are observed scale. Coinciding plots are obtained as given in the inset of Figure 2(a) indicating that the underlying mechanisms in the whole spectral range in the temperatures probed exhibit identical thermal activation. This finding is in agreement with other conducting glass-forming liquids, ion-conducting glasses, and polymers studied before.42–45

Table 1. Fit Parameters Obtained by Applying eq 4 to the Rates Corresponding to the Secondary Relaxation Processes γ and β. The VFT-Equation (eq 5) is Also Applied to the Characteristic Rates of Charge Transport, dc-Conductivity σ0, Diffusion Coefficient and Mobility for the Various Hyperbranched Polyester Amides and the Parameters are as Indicated. The Pre-Exponential Factor is Renamed in Accordance with the Quantity under Study
inline image

The dependence of the imaginary part of the complex dielectric function, ε′′, on the frequency at lower temperatures (lower than Tg) shows two relaxation mechanisms, namely, γ and β in the order of increasing temperatures/decreasing frequency for the three polymers investigated. The two secondary processes are fitted using the empirical Havriliak–Negami (HN) function given by

equation image(2)

where Δε is the dielectric strength, ε is the relaxed value of ε′, τHN is the Havriliak–Negami relaxation time, and β, γ are shape parameters. The molecular relaxation rates are related to the characteristic time obtained from eq 2 by

equation image(3)

Both relaxations show an Arrhenius-type temperature dependence (Fig. 3), which can be described by:

equation image(4)

where EA is the activation energy, ωβ,γ the relaxation rate of the process, kB the Boltzmann constant and ω the relaxation rate in the high temperature limit. The obtained values of Ea and ω by fitting eq 4 to the experimental data are given in Table 1. The γ relaxation process is fastest in case of phenyl modified polymer and exhibits the lowest activation energy (22 ± 0.5 kJ mol−1) as presented in Figure 3. This finding supports the interpretation that this mechanism is related to the relaxation of the end groups. The hydrogen bonding results in slower γ relaxation in case of the unmodified polymer, PEA-OH in comparison to PEA-Ph. β-relaxation can be attributed to the libration motion of the amide (NHCO) group bonded to the rigid phenyl ring. Figure 3 implies that both mechanisms may not be completely independent of each other as the β-relaxation has similar order to that of γ relaxation. Its activation energy is comparable for the three polymers (65–72 kJ mol−1). In addition, the Arrhenius plot of both processes in comparison to that of an aromatic hyperbranched polyurethane terminated by OH groups (with similar architecture), Ar HbPureth-OH,34 reveals that γ mechanism of both systems of OH terminal groups coincide, whereas the β mechanism is slower in case of polyurethane than in PEA. Thus, the γ relaxations can be assigned to the terminal groups.

Figure 3.

Activation plot of the characteristic rates of charge transport, ωc (half-filled symbols), and secondary relaxations (wb (open symbols) and wg (cross-filled symbols)) for the different hyperbranched polyester amides with hydroxyl (PEA-OH (squares)), phenyl (PEA-Ph (circles)), and stearate (PEA-Stearate (triangles)) terminal groups. The molecular mobility data obtained for hyperbranched polyurethane (ref.34, star symbols and dashed lines) are also included for comparison. Correlation between secondary dynamics in hyperbranched polyester amide and polyurethane with OH terminal groups is observed. The lines are fits by the Vogel–Fulcher–Tammann equation (eq. 5) for ωc and eq 4 for the rates corresponding to the secondary β and γ relaxations. The experimentally determined values of calorimetric glass transition temperature Tg are indicated by open symbols with vertical line.

The temperature dependence of charge transport quantities is approximated by Vogel–Fulcher–Tammann equation expressed as

equation image(5)

where υ = (2πτ)−1, D is a constant, T0 is the Vogel-Temperature (sometimes referred to as the ideal glass transition temperature). In contrast to ionic liquids, ωc does not scale with the calorimetric glass transition (Fig. 3). The polymeric matrix has to be considered as heterogeneous concerning charge transport. Although the H-bonded moieties contribute to electrical conduction, this is not the case for the unpolar benzene rings. This difference does not play any role in the calorimetric determination of Tg. Hence, it is to be expected that ωc does not scale with Tg. σ0 also exhibits a VFT-type temperature dependence as shown in Figure 4(a).

Figure 4.

Temperature dependence of (a) dc conductivity, σ0 (inset: σ0, vs. the characteristic rate of charge transport, ωc, for the different hyperbranched polyester amides as indicated), and (b) diffusion coefficient (open symbols) and mobility (full symbols) determined by the Einstein and Einstein–Smoluchowski equation (eq 6) (using ωc as hopping rate and λ = 0.2 nm as hopping distance). Inset: effective number density of charge carriers as functions of inverse temperature.

To gain deeper insight into the charge transport mechanism, σ0 and ωc are analyzed in further detail. The two quantities are related by an expression obtained by combining Einstein and Einstein–Smoluchowski equations. In this framework, the dc conductivity is given by

equation image(6)

where n denotes the effective number density of the ions, μ the mobility, D the diffusion coefficient, k the Boltzmann constant, λ represents the characteristic hopping length, and ωc the characteristic diffusion (charge transport) rate. It is readily observed that the empirical relation between the dc conductivity and ωc, known as the Barton–Nakajima–Namikawa (BNN) relation, is an immediate consequence of eq 6.

In recent articles,37, 43, 45, 46 it has been demonstrated that it is possible to determine diffusion coefficients from dielectric spectra using eq 6 in quantitative agreement with pulsed field gradient nuclear magnetic resonance measurements. In this approach, the characteristic hopping length is assumed to be of the order of the Pauling diameters (typically 0.2 nm) of the charge carriers. The diffusion coefficients and the corresponding mobilities (from Einstein relation) for the current hyperbranched polymers are shown in Figure 4(b). These approach enables one to separate (using eq 6) the contribution of n, the ion number density, from that of μ, the electrical mobility to the measured dc conductivity. It turns out that the VFT character of the conductivity originates from the thermal activation of the diffusion coefficient, whereas, over a broad temperature range, the number density of ions follows an Arrhenius-type temperature dependence [inset of Fig. 4(b)].

An important pointer to the dominating mechanism of charge transport is the decoupling index, Rτ, defined in terms of the structural relaxation time and the conductivity relaxation time.47 It is empirically given by: decoupling index(log) = 14.3 + log σ0(Tg). For the hyperbranched polymers investigated in this study, decoupling index of ∼5 are obtained indicating that charge transport is dominated by charge carriers that are about 5 decades faster than the structural relaxation rate of the polymer. This is typical of proton conduction as previously observed for other polymers.37 Given the chemical structure of the hyperbranched polymers investigated, it is expected that the high propensity of hydrogen bonding results into considerable ionic conductivity.


Charge transport and dipolar relaxations in hyperbranched PEAs with different terminal groups are studied by BDS. The random energy barrier model proposed by Dyre is shown to quantitatively describe the former mechanism, whereas the latter are unambiguously assigned to localized motion of the amide and the end groups of the polymers investigated. No direct correlation is found between the two processes in the polymers studied.


Financial support from the Deutsche Forschungsgemeinschaft under the DFG SPP 1191 Priority Program on Ionic Liquids is gratefully acknowledged