Dielectric Properties of Thermo-Reversible Hydrogels: The Case of a

Jul 28, 2009 - We investigated the dielectric properties of aqueous solutions of a grafted copolymer, consisting of a polysaccharide, Dextran, grafted...
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J. Phys. Chem. B 2009, 113, 11421–11428

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Dielectric Properties of Thermo-Reversible Hydrogels: The Case of a Dextran Copolymer Grafted with Poly(N-isopropylacrylamide) Giancarlo Masci† and Cesare Cametti*,‡ Dipartimento di Chimica, UniVersita´ di Roma “La Sapienza”, Rome, Italy, Dipartimento di Fisica, UniVersita´ di Roma “La Sapienza” Rome, Italy, and INFM-CNR CRS-SOFT ReceiVed: April 8, 2009; ReVised Manuscript ReceiVed: May 26, 2009

We investigated the dielectric properties of aqueous solutions of a grafted copolymer, consisting of a polysaccharide, Dextran, grafted with a thermo-sensitive polymer, poly(N-isopropylacrylamide), [pNIPAAM], over broad temperature and frequency ranges. The graft copolymers, prepared by atom-transfer radical polymerization [ATRP], form temperature-responsive materials that represent a class of self-assembled structures in water of great interest because of their potential use as drug delivery formulations and in diverse biotechnological applications. In these systems, in the dilute regime and below the lower critical solution temperature, relaxation modes corresponding to two different length-scales have been observed and analyzed in terms of ion fluctuation dielectric models specifically developed to describe the dielectric relaxation in highly charged polyion aqueous solutions. Regardless of whether the ions were produced by the ionization of the polymer chain, as in polyelectrolyte solutions, or not, as in the present case, they represent a probe at a microscopic level that is expected to reveal the structural characteristics of the system at different scales. We have identified a characteristic length associated with the size of the polymer coil in the dilute regime and a length due to fixed cross-links, where ions are partially localized by the local profile of the Coulombic field. These lengths are in reasonable agreement with analogous lengths derived from structural information and from the hydrodynamic radius of the polymer coils, measured by means of a dynamic light-scattering technique. 1. Introduction Stimuli-responsive hydrogels are a class of materials that have received in the past few years increasing attention because of a wide variety of applications in biomedical fields, including drug delivery and tissue engineering.1-3 Usually, chemically crosslinked gels made up of stimuli-responsive polymers are commonly used for these biotechnological applications, even though they can have some drawbacks, such as the presence of unreacted cross-linking reagents that can lead to high toxicity.4 Physical gels derived from B-A-B triblock copolymers and graft copolymers have been extensively investigated because of the noncovalent nature of their cross-links and their potential reversibility. At moderate to high polymer concentrations, physical cross-links can form when part of the polymer becomes incompatible with the solvent due to a proper stimulus (temperature, pH, ionic strength).1,5-7 Among thermo-sensitive polymers, poly(N-isopropylacrylamide) [pNIPAAM] has been extensively investigated. Below the lower critical solution temperature (LCST, about 32 °C), the polymer is soluble, whereas when the temperature is raised above the LSCT, pNIPAAM chains become hydrophobic, and a phase separation takes place.8,9 Graft, triblock, and random copolymers containing pNIPAAM have been widely used to prepare reversible, temperature-sensitive, physical hydrogels.10-16 Belonging to these thermo-sensitive polymers, polysaccharides grafted with pNIPAAM represent a class of very interesting materials because of their biotechnological applications. Thermosensitive physically gelating derivatives of polysaccharides can * To whom correspondence should be addressed. E-mail: cesare.cametti@ roma1.infn.it. † Dipartimento di Chimica. ‡ Dipartimento di Fisica.

be obtained by grafting with pNIPAAM. At room temperature, both the polysaccharide and pNIPAAM polymers are hydrophilic, and the grafted chains are soluble in water. By raising the temperature, pNIPAAM chains collapse and are segregated in hydrophobic domains that constitute the cross-linking points of the hydrogel, whereas the polysaccharide chains, between insertion points of pNIPAAM, act as elastic junctions (see the sketch in Figure 1). Usually, ceric ion initiation, Fenton’s reagent, and γ-radiation were used to graft synthetic polymers onto polysaccharides. Recently, we have demonstrated that atom transfer radical polymerization (ATRP),17,18 a controlled/“living” radical polymerization technique that minimizes chain transfer and termination and allows control of the molecular weight and polydispersity, can be used as a versatile method for homogeneous grafting of underivatized polysaccharides in mild conditions, with a variety of vinyl monomers, including NIPAAM.19 By this method, thermosensitive derivatives of dextran, an essentially linear polysaccharide with R(1-6) linked D-glucopyranosyl residues, with grafted pNIPAAM chains (Dex-gpNIPAAM) have been prepared.19,20 At low concentrations, Dex-g-pNIPAAM forms polymeric micelles upon increasing the temperature. In this work, we have investigated the radiowave dielectric properties of aqueous solutions of Dex-g-pNIPAAM at moderate or high concentrations. Measurements have been carried out over an extended temperature range, from 10 to 50 °C, crossing the LCST and at different polymer concentrations, from the dilute to concentrated regime. In these conditions, Dex-gpNIPAAM forms gels by increasing the temperature above the LCST (Figure 1). Dielectric spectroscopy, being especially sensitive to the heterogeneity of the system, can provide insights into the

10.1021/jp9032468 CCC: $40.75  2009 American Chemical Society Published on Web 07/28/2009

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Figure 1. Reversible formation of Dex-g-pNIPAAM hydrogel by changing the temperature above or below the LCST.

structure and the electrical properties at a molecular level in studying molecular relaxations in polymers. Since dielectric spectroscopy in the frequency domain covers a broad frequency range, in our case, from 100 Hz to 2 GHz, this technique allows measurements of different relaxation processes simultaneously, giving useful information that includes, in addition to the electrical properties of each domain in the heterogeneous solution, the movement of the whole polymer chain or of its side chains and the diffusion of ions over different lengths up to the typical size of different regions that may characterize the whole solution. By means of an appropriate deconvolution of the whole dielectric response, we have observed two different contributions falling at frequencies on the order of 1-10 and 100 MHz, respectively. These dielectric relaxations are characterized by dielectric strengths ∆ε and relaxation frequencies ν that depend, in a rather complex way, on the polymer concentration and the temperature. In analogy with dielectric models that consider the electric polarization of counterions induced by an external electric field, the observed dielectric relaxations are ascribed to diffusion processes of ions bound to the Coulombic potential around the polymer by crossing through the cross-linking points of the polymer chain, as far as the relaxation at higher frequencies is concerned, and over the whole polymer domain, as far as the relaxation at lower frequencies is concerned. 2. Experimental Section 2.1. Material. Dextran (T-70, Amersham-Pharmacia, Uppsala, Sweden, number average molecular weights 70 000 g/mol) was used as received. N-Isopropylacrylamide (NIPAAM, Aldrich) was recrystallized from hexane. 2.2. Grafting of Dextran with pNIPAAM. pNIPAAMgrafted dextran [Dex-g-pNIPAAM] was prepared by means of a “grafting from” approach, as already reported elsewhere.19,20 Dextran macroinitiator with a degree of substitution DS ) 10% (DS, the number of initiating CP groups for 100 glucose units of dextran) was prepared. pNIPAAM chains with an average degree of polymerization Xn ) 40 were grafted. Grafted chains had a low polydispersity index (1.24). The grafted copolymer will be named Dex10-g-pNIPAAM40. 2.3. Rheological Characterization. Rheological measurements were carried out by means of a Bohlin CS10 stresscontrolled rheometer using a concentric cylinder measuring system (C14). About 2 mL of polymer solution was charged, and a thin layer of mineral oil was added above the reaction mixture to prevent evaporation of the solvent. Measurements were made between 25 and 50 °C ((0.1 °C). Creep tests were performed to ascertain the absence of any slip effect. Storage and loss moduli, G′ and G′′, respectively, were determined in the oscillatory regime at a frequency of 1 Hz and a nominal strain of 5 × 10-3. The evolution of the storage G′ and loss G′′ moduli of a Dex10-g-pNIPAAM40 aqueous solution at polymer concentra-

Figure 2. Temperature dependence of storage G′ (•) and loss G′′ (O) moduli of Dex10-g-pNIPAAM40 polymer in aqueous solution at a frequency of 1 Hz. The polymer concentration is Cp ) 50 mg/mL.

tion Cp ) 50 mg/mL is shown in Figure 2. At temperatures below LCST, we observe values of G′ lower than G′′, which is typical of the liquid state with dominant viscous properties and energy more dissipated than stored. By crossing LCST, a strong increase in G′ and G′′ was observed, and G′ became higher than G′′, indicating the formation of a gel. The gelation temperature determined as the crossover point of G′ and G′′ in their temperature dependence was about 42 °C.21 The same value was obtained using the most general method based on the observation that, at the gel point, tan δ ) G′′/G′ is independent of the frequency (data not reported).22 The high value of LCST measured for Dex10-g-pNIPAAM40 with respect to the usually reported value for pNIPAAM polymer (T ) 32 °C) is due both to the low molecular weight of low polydisperse pNIPAAM and to the influence of hydrophilic dextran.23 2.4. Radiowave Dielectric Measurements. The dielectric and conductometric spectra of Dex10-g-pNIPAAM40 polymers in aqueous solutions at different temperatures have been measured in the frequency range from 1 kHz to 2 GHz by means of two radio frequency impedance analyzers, Hewlett-Packard models 4294A (in the frequency range from 1 kHz to 10 MHz) and 4291A (in the frequency range from 1 MHz to 2 GHz). Details of the dielectric cell and the calibration procedure have been reported elsewhere.24,25 The samples investigated at three different concentrations of 30, 50, and 70 mg/mL were heated from 10 to 50 °C at a rate of 0.1 °C/min in a closed environment. The temperature-induced formation of the hydrogel at a LCST of pNIPAAM was recorded by the appearance of a change in the slope of temperature dependence of the electrical conductivity (data not shown). In the systems investigated, the polymer concentration in terms of fractional volume occupied within the sample is relatively small, on the order of Φ ) 0.05. What’s more, the electrical conductivity of the suspension, thanks to the ionic residues in the reaction mixture of the polymer solutions investigated, is relatively high, ranging between 0.1 and about 0.3 mho/m, depending on polymer concentration and the

Thermo-Reversible Hydrogels

J. Phys. Chem. B, Vol. 113, No. 33, 2009 11423 As can be seen, the electrode polarization effect dominates the low-frequency region of the spectrum, where the apparent permittivity ε′(ω) scales with the frequency as ω-R (with R ) 1.63, in this case) (Figure 3, upper panel). Analogously, the conductivity loss exhibits a simple ω-1 frequency dependence over almost the whole frequency range of the spectrum, and deviations indicating a dielectric loss contribution to the total loss occur only at frequencies higher than 5 × 106 Hz (Figure 3, bottom panel). According to that stated above, the whole dielectric spectra have been described on the basis of the superposition of two Cole-Cole relaxation functions26,27 modified by adding a further Debye relaxation to take into account the contribution of the dielectric response at higher frequencies (orientational polarization of the aqueous phase) and by adding the contribution of a constant-phase-angle (CPA) element to take into account the contribution of the electrode polarization at lower frequencies. The complete relaxation function for the complex dielectric constant, ε*(ω), reads

A(iω)-R Figure 3. Upper panel: the apparent permittivity, ε′(ω), of the Dex10-g-pNIPAAM40 polymer solution as a function of frequency at a temperature of 24 °C. In the low frequency tail of the frequency range investigated, the electrode polarization effect dominates, and the permittivity, ε′(ω), obeys the scaling law, ∼ω-R. The inset shows, in an extended linear scale, the presence of a dielectric relaxation described by a Cole-Cole relaxation function (full line). Bottom panel: the total dielectric loss σ/ε0ω of the Dex10-g-pNIPAAM40 polymer solution as a function of frequency, at a temperature of 24 °C. In the low-frequency region, the scaling dependence ∼1/ω is obeyed. The inset shows deviation from the scaling, ∼1/ω, being the print of a dielectric relaxation.

temperature. These two factors contribute to produce a relatively low dielectric response, masked by two concomitant effects. The first one, in the low frequency tail of the frequency window investigated, is due to the electrode polarization effect, which acts, from a dielectric point of view, as a complex impedance Zp(ω) ) A(iω)-R in series with the impedance associated with the sample. The second factor is associated with the small value of the intrinsic phase angle θ for a conducting solution, which decreases with the increase in the electrical conductivity, σ, and decrease in the angular frequency, ω, according to tan θ ) ωε0ε/ σ. For a permittivity ε on the order of ε = 102 and an electrical conductivity σ = 10-1 mho/m, as in the present case, the phase angle is less than 10-4 rad, making an accurate measurement of the dielectric contribution quite difficult. Moreover, the conductivity of the whole system will contain two contributions, one from the d.c. conductivity, σ0, and the second one from the dielectric loss, εdiel ′′ (ω), according to ″ (ω) σ(ω) ) σ0 + ε0ωεdiel

(1)

and, in the case of high conductivity, the first term may dominate the second one at lower frequencies. A typical dielectric spectrum of a Dex10-g-pNIPAAM40 polymer solution at a temperature of 24 °C in the whole frequency range investigated, from 100 Hz to 2 GHz, is shown in Figure 3.

″ (ω) ) ε*(ω) ) ε′(ω) - iεtot ∆εW σ0 ∆εj + ε∞ + + + 1 + iωτW iωε0 1 + (iωτj)βj j)1,2 (2)

∑

′′ (ω) are the permittivity and the total where ε′(ω) and εtot dielectric loss, respectively; and ∆εj, νj ) 1/2πτj and βj, (j ) 1, 2) are the dielectric strengths (the dielectric increments), the relaxation frequencies, and the parameters that represent the broadness of the distribution of the relaxation frequencies, respectively. ε∞ is the high-frequency limit of the permittivity ε′(ω), and ∆εW and τW are the dielectric increment and the relaxation time of the aqueous phase relaxation. Finally, σ0 is the dc electrical conductivity of the polymer solution. The electrode polarization effect is taken explicitly into account by the term A(iω)-R, where A and R are adjustable parameters derived from the low-frequency part of the frequency window investigated by means of the fitting procedure employed. In this case, the real part of eq 2 was fitted to the experimental data because of the different order of magnitude between the apparent ′′ . permittivity ε′ and the total dielectric loss εtot In the following, we present and discuss separately typical dielectric spectra of Dextran aqueous solutions; of pNIPAAM aqueous solution; and finally, of Dex10-g-pNIPAAM40 aqueous solutions. 3. Results 3.1. The Dielectric Behavior of Dextran Aqueous Solutions. Dielectric spectra of a Dextran aqueous solution (polymer concentration 13 mg/mL) at two temperatures (24 and 46 °C) in the frequency range from 10 kHz to 2 GHz are shown in Figure 4. In this frequency range, the permittivity ε′(ω) is largely independentsat least in the frequency range investigatedsof frequency, as one should expect for nonionic polymer solutions. The behavior of the permittivity ε′(ω) with the increase in the temperature is shown in the inset of Figure 4. This behavior reflects the one of pure water. Dextran in aqueous solution does not alter, to a first approximation, the dielectric response of the water phase, the interactions being governed by the excluded volume of the solution. 3.2. The Dielectric Behavior of pNIPAAM Aqueous Solutions. Thermosensitive properties of pNIPAAM were clearly confirmed by dielectric spectroscopy measurements that show

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Figure 4. The permittivity of Dextran aqueous solution as a function of frequency at a polymer concentration of C ) 13 mg/mL, for two different temperatures (below, 24 °C, and above, 46 °C; the lower critical solution temperature (LCST) observed in the Dex-g-pNIPAAM polymer solution and in pNIPAAM solution). The inset shows the permittivity as a function of temperature.

Figure 6. The dielectric spectrum of Dex10-g-pNIPAAM40 aqueous solutions as a function of frequency at temperatures of 20 °C (below the LCST) and 48 °C (above the LCST) for three different polymer concentrations (30, 50, and 70 mg/mL). The overall dielectric response of the polymer solutions at concentrations of 30 and 50 mg/mL is due to two different contiguous dielectric relaxations. The deconvolution of the spectrum results in two different Cole-Cole relaxation processes, shown as a continuous line in the figure. The two arrows mark the relaxation frequencies of the two processes. At the higher concentration (70 mg/mL), the dielectric response is characterized by a single Cole-Cole relaxation process. The arrow marks the relaxation frequency. Figure 5. Dielectric spectra of pNIPAAM polymer solutions as a function of frequency at two different temperatures below (T ) 16 °C, upper panel) and above (T ) 46 °C, bottom panel) the LCST. The dotted line represents the response of the aqueous phase in the absence of the polymer dielectric dispersion.

the presence of a dielectric dispersion whose strength (∆ε) progressively decreases with an increase in the temperature. Figure 5 shows the dielectric response of pNIPAAM polymer solution (at a concentration of 37 mg/mL) at two typical temperatures below (T ) 16 °C) and above (T ) 46 °C) the LCST. As can be seen, when the polymer is stable in the solution (below LCST), a well-defined dielectric dispersion occurs. This dielectric dispersion progressively reduces its strength (∆ε) with the increase in the temperature until it disappears at temperatures higher than the LCST. In this case, the phase separation induces a dielectric response that resembles the one of the pure water (bottom panel of Figure 5). 3.3. The Dielectric Properties of Dextran-GraftedpNIPAAM Aqueous Solutions. The dielectric spectra of Dex10-g-pNIPAAM40 aqueous solutions at the three different polymer concentrations investigated (30, 50, and 70 mg/mL, respectively) have been measured in the frequency range from 1 kHz to 2 GHz at different temperatures crossing the LCST value, from 10 to 50 °C. Typical dielectric spectra in the high-frequency range, for each of the three concentrations investigated and corrected for the electrode polarization effect, are shown in Figure 6 at two different temperatures (20 and 48 °C) below and above the LCST of pNIPAAM in water.

These spectra present a rather complex structure, depending on the temperature and the polymer concentration. Here, we will summarize their main characteristics. At the two lower polymer concentrations (30 and 50 mg/ mL), the spectra result from the partial overlap of two adjacent relaxation processes. Consequently, the spectra have been analyzed as the sum of two Cole-Cole relaxation functions (j ) 1, 2 in eq 2). At the higher polymer concentration investigated (70 mg/ mL), for each temperature, the spectra can be more conveniently described by a single Cole-Cole relaxation function (j ) 1 in eq 2). The dielectric relaxations are described by means of the ColeCole function, which is one of the most frequently used relationships to take into account the dielectric relaxations of polymer solutions. Other relaxation functions, such as the Havriliak-Negami27 function, require a larger number of parameters, and we prefer to keep this number as small as possible. The choice of describing the dielectric spectra with a different number of adjacent relaxations in dependence on the polymer concentration deserves a further comment. Since in the analysis of the data, the two different dielectric models involve a different number of parameters (three for a single Cole-Cole function and six for two adjacent Cole-Cole functions), it is important to verify on the basis of an appropriate statistical test whether a model is statistically better than the other to describe the same set of m observations only by virtue of the increased number

Thermo-Reversible Hydrogels

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Figure 7. The dielectric increments ∆ε1 (at lower frequencies) and ∆ε2 (at higher frequencies) of the relaxation processes observed in Dex10-g-pNIPAAM40 aqueous solution as a function of the temperature, for three different polymer concentrations (30, 50, 70 mg/mL). The dotted line is the sum of the two dielectric strengths, ∆ε1 and ∆ε2.

of parameters. To decide this, the error sum, S, for each model (here denoted as SA and SB) m

S)

∑ (Xobs - Xcalc )2

i)1,m

i

i

(3)

can be used to calculate the function

f(m, p, q) ) (SA - SB)(m - p)/SB(q - p)

(4)

where p and q are the free parameters of models A and B, respectively, which is assumed to be distributed according to the F statistical function F(q-p, m-q, R), with (q-p) and (m-q) degrees of freedom. The values of f(m, p, q) at 95% confidence level, obtained in the two above data analysis, despite the scattering of the experimental value and the different number of free parameters, attest that the two lower concentrations (30 and 50 mg/mL) for each temperature investigated require the presence of two adjacent relaxation processes, whereas the higher concentration (70 mg/mL) requires the presence for each temperature of only one relaxation process. Consequently, the fitting procedure, using the nonlinear leastsquares method, was carried out in the frequency region that is not influenced by the electrode polarization effect (or this effect was corrected when necessary by the procedure previously described) using one or two adjacent Cole-Cole relaxation functions, depending on the polymer concentration. The obtained best-fitting parameters ∆ε1, ∆ε2 and ν1, ν2 for the two lower concentration (30 and 50 mg/mL) and ∆ε and ν at the higher concentration (70 mg/mL) are shown as a function of temperature in Figures 7 and 8, respectively. The parameters βj of the two adjacent relaxation regions, which take into account the spread of the relaxation times, are on the order of 0.10-0.15, slightly depending on the temperature and the polymer concentration.

Figure 8. The relaxation frequency ν1 (at lower frequencies) and ν2 (at higher frequencies) of the relaxation processes observed in Dex10-gpNIPAAM40 aqueous solution at three different polymer concentrations (30, 50, and 70 mg/mL) as a function of the temperature.

As can be seen, the two dielectric dispersions, characterized by the dielectric strength, ∆ε, and the relaxation frequency, ν, behave differently as a function of temperature. For the two solutions at lower polymer concentrations (30 and 50 mg/mL), the dielectric increment ∆ε1 at lower frequency increases approximatively linearly with temperature, showing two welldefined different slopes, the intersecting temperature being at about 41-42 °C, which is in very good agreement with the LCST value measured by rheology (Figure 2). In the same temperature interval, the dielectric increment ∆ε2 of higher frequency dielectric process decreases approximately linearly within the experimental uncertainties without, or with a more or less change in its slope, close to the LCST (Figure 7). For the solution at higher concentration (70 mg/mL), the dielectric increment ∆ε is approximately constant up to the LCST, and then it shows a marked increase. The behavior of the relaxation frequencies of the two processes is shown in Figure 8. As can be seen, in the case of the two lower concentrations (30 and 50 mg/mL), the process at lower frequency is characterized by a relaxation frequency, ν1, on the order of 3-8 MHz, approximately constant in the whole frequency range investigated without any appreciable change in the slope crossing the LCST. In contrast, the process at higher frequency is characterized by a relaxation frequency, ν2, on the order of 80-100 MHz, with a marked decrease toward lower values at the LCST. This picture is different in the case of the solution at higher polymer concentration (70 mg/mL), where the process at higher frequency has, to a first approximation, disappeared and only the process at lower frequency persists. In other words, the process at higher frequencies shifts toward the one at lower frequencies, as the polymer concentration increases. 4. Discussion After this phenomenological rather complex description of the observed dielectric behavior, we will try to give a more quantitative interpretation of the observed dielectric parameters

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on the basis of some physicochemical characteristics of the system investigated. First of all, we evaluate the critical concentration, c*, that defines the transition from the dilute to the concentrated regime. It can be defined as the concentration at which coils start to overlap; that is, the volume of the solution, Vsol, equals the volume of the coils, Vcoils. If the simple coil is approximated by a sphere with a radius equal to the hydrodynamic radius RH, the above stated condition yields

c* )

Mw M ) 3 n N 4/3πRH N 4/3πRH3

(5)

where n ) M/Mw is the number of moles, M is the mass of the polymer expressed in grams, Mw is the molecular weight of the polymer, and N is the Avogadro number. Considering for Dex10-g-pNIPAAM40 polymer a molecular weight of Mw ) 2.66 × 105 g/mol and a hydrodynamic radius of RH ) 12.5 nm, as determined by dynamic light-scattering measurements in dilute polymer solution, we obtain a value of c* ) 54 mg/ mL. According to this value, the two lower concentrations (30 and 50 mg/mL) refer to dilute solutions (50 mg/mL is only a little bit lower than the value c*), whereas the higher concentration (70 mg/mL) refers to a concentrated solution. It follows that the different dielectric behavior as a function of the polymer concentration reflects, at least in part, the different concentration regimes of the samples investigated. In the frequency range investigated, dielectric relaxation is expected to reveal the ion fluctuations induced by the external electric field, along with some typical lengths that are characteristic of the local profile of the Coulombic potential around and along the polymer chain.28 The presence of two contiguous dielectric dispersions indicates that the relaxation occurs on two different time scales and allows us to identify two different characteristic lengths of the system. We will consider here a dielectric model originally developed to describe the dielectric behavior of aqueous linear polyelectrolyte solutions on the basis of the fluctuation of the counterion atmosphere around the polyion chain.29-32 According to this model, the polyion is represented by a sequence of identical subunits surrounded by a fraction of the total counterions, which are closely associated with the subunit itself. In the presence of an external electric field, this ion distribution is perturbed, giving rise to an induced dipole moment roughly proportional to the square of the average extension of the polymer chain where ions are bound. The relaxation of the resulting polarization occurs on two different time scales. On the shorter one, ion fluctuation is confined within the single subunit because of potential barriers opposing the ion diffusion. On the longer time scale, ions are able to overcome these barriers, and ion distribution tends to be uniform on the whole polyion chain. These ion distribution mechanisms give rise to a dielectric response characterized by two dielectric relaxation processes at higher and lower frequencies, respectively. Models based on the counterion polarization have been extensively employed in the past to take into account the dielectric behavior of a wide class of synthetic and natural polyelectrolyte systems33,34 and a generally good agreement with experimental values has been found. Despite the origin of the model that invokes the counterion polarization has put forward in light of the dielectric behavior of highly charged aqueous solutions,29,30 the same basic mechanism can be considered here in the case of graft copolymers,

since the role played by the counterions in the above stated model can be assigned to ions present in the system (ions derived from ionic residues) whenever some kind of potential barriers exist along the polymer chain. Ions freely mobile over lengths associated with the typical size of the polymer coil (at concentrations c < c*) produce the polarization process occurring at the lower frequencies (at about 5-10 MHz) characterized by a relaxation frequency, ν1, given by

ν1 =

DL 2πL2

(6)

where L and DL are the linear dimension of the coil and the diffusion coefficient of ions, respectively. In contrast, the component at higher frequencies (at about 100 MHz) of the observed relaxation at c < c* should correspond to the counterion fluctuation along the polymer chains by crossing through the insertion points of pNIPAAM chains. In this case, the relaxation frequency, ν2, is analogously given by

ν2 =

D Lf 2πLf2

(7)

where Lf and DLf are, again, the fluctuation length and the diffusion coefficient of ions. The diffusion distance Lf reflects in this case the width of the Coulombic potential valley between effective cross-linking points, and DLf, the diffusion coefficients of this polarization mechanism. These two above-stated polarization mechanisms, neglecting correlations between the associated ions distributed along the chain, result in two dielectric strengths, the former at low frequency, given by

NL(ze)2CLL2 ∆ε1 ) 36ε0KbT

(8)

and the latter, at higher frequency, given by

∆ε2 )

NLf(ze)2CLfLf2 36ε0KbT

(9)

where NL and NLf are the average fractions of associated ions per each polymer chain of size L and per each subunit of length Lf, respectively; ze is the charge of the ion; CL and CLf are the numerical concentration of the polymers and of the subunits per unit volume, respectively; KbT is the thermal energy, and ε0 is the dielectric constant of free space. For full details, one should consult the original works by Mandel.29 Although the theory is rather crude and the introduction of the potential barriers needs to be considered in a more rigorous way, is was found adequate for understanding most of the details of the dielectric behavior of linear polyelectrolytes in a rather quantitative manner. Since from the analysis of the dielectric spectra the dielectric strength, ∆ε, and the relaxation frequency, ν, have been properly evaluated, the model parameters, that is, the lengthes L and Lf and the ion concentrations NL and NLf, can be deduced from eqs 6-9.

Thermo-Reversible Hydrogels

Figure 9. Dependence of the typical size, L, as a function of temperature for polymer solutions at different concentration regimes: a dilute regime, 30 and 50 mg/mL, and a concentrated regime, 70 mg/ mL. (O), 30 mg/mL; (0), 50 mg/mL; (∆), 70 mg/mL.

Figure 10. Dependence of the typical size, Lf, as a function of temperature for polymer solutions at different concentrations (30 and 50 mg/mL) in the dilute regime: (O), 30 mg/mL; (0), 50 mg/mL.

In Figures 9 and 10, the characteristic lengths L and Lf, calculated on the basis of eqs 6 and 7 taking into account the measured values of the relaxation frequencies ν1 and ν2, are shown as a function of temperature for the three polymer concentrations investigated. In these calculations, we have assumed a diffusion coefficient on the order of DL ) DLf )10-6 cm2/s (with a temperature dependence similar to the one of the Na+ ion in a free medium). As far as the length, L, is concerned, this value is in reasonable agreement with the ones obtained for the hydrodynamic radius of the polymer coil measured by dynamic lightscattering techniques. In the case of a polymer at a concentration C ) 10 mg/mL (dilute solution), at room temperature, the measured hydrodynamic size is 2RH ) 25 nm. These findings suggest assigning the low-frequency dispersion to ion fluctuations over the whole polymer coil. The length Lf, which originates the high-frequency dispersion, is associated with the contiguous insertion points of pNIPAAM. As a matter of fact, the contour length of glucose repeating unit of dextran is 0.45 nm.37 Dex10-g-pNIPAAM40 polymer has a degree of substitution, DS ) 10%, which means that a pNIPAAM chain is grafted every 10 glucose residues. Therefore, considering a fully extended dextran chain, the average distance between insertion points of pNIPAAM is 4.5 nm. This value is in very good agreement with the ones deduced from the dielectric dispersion in the samples at a concentration below c* and for temperatures below LCST, with individual coils present in the solution (Figure 10). Above the LCST, the Lf markedly increases toward the values observed in the sample at higher concentration (70 mg/mL). This progressive shift of the smaller typical length as the concentration is increased toward the unique higher value

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Figure 11. The polymer concentration dependence of the typical lengths, L and Lf, at three different temperatures: (•), T ) 42 °C; (9), T ) 20 °C.

corresponding to the higher concentration is shown in Figure 11 for two different temperatures, below, close to, and above the LCST. At high concentration and above the LCST, both Lf and L converge to similar values that could be representative of the average distance between the cross-linking points. These crosslinking points will be due to the formation of pNIPAAM hydrophobic domains above the LCST for all the concentrations investigated. Above c* and below LCST, they are due to the topological entanglement points between overlapped chains. Equations 8 and 9 allow the fluctuating ion concentration, NL and NLf, along the two typical lengths, L and Lf, to be determined, as well. As far as the high-frequency relaxation is concerned, ions fluctuate along subunits bounded by a potential barrier at both of their ends, which arises from a variation in the local conformation at the junction point of the pNIPAAM polymer of two neighboring subunits. Taking into account that the number-average degree of polymerization of Dextram is 432 and DS ) 10%, on average, there are about 43 glucose decamer subunits per Dex10-g-pNIPAAM40 polymer. This value furnishes for the concentrations CLf values of 2.9 × 1024 and 4.6 × 1024 subunit/m3 for samples at the two lower concentrations. By using for the length Lf the values deduced from the relaxation frequency ν2 (eq 7), eq 9 yields for temperatures lower than the LCST a value of NLf of 6 × 10-4 mol/L and of 10-3 mol/L for samples 30 and 50 mg/mL, respectively. In these calculations, we have employed the measured values of the dielectric increment ∆ε2 as a function of the temperature. These values are very close to one another when normalized to the polymer concentration, suggesting that there is no polymer concentration dependence, as requested by a dielectric relaxation mechanism that involves only the local structure of the polymer chain. The same overall agreement has been found as far as the lowfrequency dielectric dispersion is concerned. In this case, being interested in the whole polymer chain, the distribution of ions occurs on a longer time scale, and the ions are able to overcome the potential barriers bounding each subunit. In this case, too, taking into account the numerical concentration of the polymer coils in the dilute regime for the two polymers at lower concentrations (Cp ) 6.8 × 1022 and 1.1 × 1023 coil per m3, corresponding to the concentrations of 30 and 50 mg/mL, respectively) and using for the length L the values deduced from the relaxation frequency, ν1 (eq 6), eq 8, considering the measured values for the low-frequency dielectric strength ∆ε1, furnishes values of NL on the order of 7 × 10-4 and 4 × 10-3 mol/L for the two concentrations, respectively, slightly decreasing with the increase in the temperature. These values deserve a further comment. The samples investigated show an electrical conductivity σ on the order of 10-2-10-1

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mho/m, depending on the temperature and polymer concentration. With the assumption that the dc conductivity is given to a first approximation only by the ions present in the solution (neglecting the contribution of the polymer), their concentration can be easily estimated if one assumes for the ion mobility the value, for example, of Na+ ions. We find values for the ion concentrations that are in reasonable agreement with the ones deduced from the dielectric increments of the two dielectric relaxations. At higher temperature above the LCST and in the concentrated regime, the above stated dielectric model fails, since chain-chain interactions prevail, losing the polymer coil its individuality. However, some further comments can be made. Under these conditions, the high-frequency relaxation disappears, and only the low-frequency effect persists. This could suggest a deep restructuring of the whole system that, due to the topological entanglement, presents at a large length-scale a more uniform structure. Moreover, the increase in the dielectric increment ∆ε1 (or ∆ε, for the sample at 70 mg/mL) at temperature higher than the LCST might be due to the formation of pNIPAAM domains because of hydrophobic interactions. These findings altogether show a good degree of selfconsistency which supports the use of models based on ion fluctuations and developed specifically to take into account the dielectric relaxation of highly charged polyion solutions. These models can, as well, describe the experimental data in the dilute regime, at least in the temperature interval up to LCST, concerning the rather different system, such as Dex-gpNIPAAM polymer solution. 5. Conclusions Although it is not possible to completely reject other dielectric relaxation mechanisms, such as a Maxwell-Wagner interfacial processes35 or polarizations due to side chain effects26,36 or even relaxations associated with the presence of bound water close to the polymer chains, the whole of the findings presented in this paper provides evidence toward a process involving ion fluctuation on two different length scales characteristic of the polymer chain investigated. They are the average size of the polymer coil in the dilute regime and below the LCST and the subunits along the polymer chain, consisting of sections of dextran molecules disjoined by points of pNIPAAM insertion. Because of potential barrier distribution that arises between neighboring subunits, under the influence of an external electric field, polarization of ions occurs within confined segments of the polymer chain, and only on a longer time scale, the distribution of ions concerns the complete polymer domain. The dielectric spectra presented in this work clearly evidence the presence of two contiguous dispersions whose characteristic parameterssthe dielectric strength, ∆ε, and the relaxation frequency, νscan be justified in the light of ion fluctuation models. We have associated the observed dielectric relaxations occurring at lower frequencies with the typical size of the polymer coil in the dilute regime and the one at higher

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