Contribution of Linear Guest and Structural Pendant Chains to

Article pubs.acs.org/Macromolecules

Contribution of Linear Guest and Structural Pendant Chains to Relaxational Dynamics in Model Polymer Networks Probed by TimeDomain 1H NMR F. Campise,† L. E. Roth,‡ R. H. Acosta,† M. A. Villar,‡ E. M. Vallés,‡ G. A. Monti,*,† and D. A. Vega*,§ †

FAMAF, Universidad Nacional de Córdoba, IFEG-CONICET, Córdoba, Argentina Deparment of Chemical Engineering, Planta Piloto de Ingeniería Química, and §Deparment of Physics, Instituto de Física del Sur (IFISUR), Universidad Nacional del Sur, CONICET, Bahía Blanca, Argentina

‡

ABSTRACT: A series of end-linked polymer networks with varying contents of linear guest chains were investigated through swelling and time-domain NMR temperature dependent experiments. Taking advantage of the thermorheological simplicity of polydimethylsiloxane polymers, time−temperature superposition (TTS) was employed to expand the characteristic time scales of NMR exploration by about 2 orders of magnitude. A comparison between swelling data and tube model predictions reveals that NMR captures the dominant features of the equilibrium and dynamic properties of defects trapped in slightly cross-linked, entanglementdominated polymer networks. As high-temperature experiments ensures a complete relaxation of the guest linear chains on the millisecond time scale of the NMR experiments, an accurate description of the network architecture can be provided. Contents of guest chains determined by NMR were found to agree within a 1 wt % accuracy with data of swelling experiments.

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INTRODUCTION Model polymer networks are ideal candidates to assess information on the dynamics of polymer defects and their influence on the elasticity of the system. In an ideal scenario, for stoichiometrically balanced systems all polymer chains are cross-linked to the matrix. However, due to a lack of absolute control over the synthesis procedure, a fraction of polymer chains either will be attached to the network by one chain-end only, constituting pendant material or become unattached “soluble” material.1,2 These defects play a crucial role in controlling the dissipative dynamics of polymer networks and therefore have an enormous technological interest, for example in the design of dynamic vibration absorbers or in the tire industry.3 Usually, information on network architecture is indirectly inferred through viscoelastic and swelling experiments, which are combined with different theories that describe equilibrium and dynamic properties.1,2,4,5 However, a multiplicity of factors, including trapped entanglements, cross-linker fluctuations, non-Gaussian contributions, and a myriad of relaxation mechanisms, make the connection among network architecture and response functions quite difficult. Hence, it is very important to develop reliable characterization methods that allow accurate determination of the network parameters that control their response. NMR is a suitable technique to study complex polymer dynamics. On a microscopic level, the restriction in mobility of polymer chains gives rise to residual dipolar couplings (RDCs) which provide information on segmental dynamics in terms of © XXXX American Chemical Society

an orientation autocorrelation function with respect to the external magnetic field.6,7 RDCs can be probed by measuring multiple quantum coherence; in particular, double quantum (DQ) NMR of pendant chains at low concentrations in poly(dimethylsiloxane) (PDMS) networks correlates with elastic shear modulus, determined by changes in viscoelastic relaxation times.1 Unrelaxed topological constraints involving pendant material impose a nonzero average dipolar coupling that contributes to the solid-like behavior of the NMR response. For instance, a significant increase in the RDCs obtained by DQ NMR was observed as a consequence of the transiently trapped entanglements.8 In particular, the influence of defects in polymer elasticity can be determined through comparison of DQ NMR and swelling experiments with the recursive method proposed by Miller and Macosko.4,5 On the other hand, the fraction of chains that contribute to elasticity at the typical NMR time scales (∼milliseconds) can be determined by direct inspection of the decay of the transverse magnetization in a multipulse sequence, such as in a Carr− Purcell−Meiboom−Gill (CPMG). In this experiment, the elastic fraction composed by cross-linked chains and trapped defects is described with the Andersson−Weiss model. Relaxed pendant material and free chains decay with the typical exponential behavior associated with liquid-like compoReceived: August 18, 2015 Revised: December 4, 2015

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DOI: 10.1021/acs.macromol.5b01806 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Table 1. Networks Prepared with Different Amount of Linear Guest Chains (WB0) of Different Molecular Weightsa network

Mn [kg/mol]

Mw/Mn

r

WB0 [g/g]

Ws [g/g]

ϕ

Wsw g [g/g]

B2-00 B2-B0,1-20 B2-B0,2-05 B2-B0,2-10 B2-B0,2-20

47.8 97.2 97.2 97.2

1.07 1.24 1.24 1.24

0.998 1.005 1.035 1.005 1.015

0.196 0.049 0.099 0.196

0.004 0.183 0.031 0.057 0.132

0.25 0.22 0.24 0.23 0.20

0.01 0.02 0.04 0.06

Ws: amount of soluble material extracted through swelling experiments. ϕ: degree of swelling in toluene. Wsw g : fraction of guest chains that remain in the network after swelling experiments.

a

nents.9−11 There is a linear dependence of the overall magnetization with the elastic chain fraction and the contribution of defects that renders the determination of network parameters quite robust. In this work, network architecture and physical entanglement dynamics in slightly cross-linked, entanglement-dominated polymer networks were investigated by means of time-domain NMR (TD-NMR) experiments.

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guest B0 chains within the network. After reaction was completed, networks were subject to differential extraction of solubles using toluene as solvent.12 The extraction of solubles was carried out at room temperature for 1 month, and solvent was replaced every 3 days. Following extraction, samples were weighed, and the degree of swelling was obtained. Samples were then dried under vacuum at 40 °C until complete solvent removal was achieved. Dry networks were weighed again, and weight fraction of solubles (Ws) and equilibrium degree of swelling ϕ (volume fraction) were calculated (see Table 1). NMR Experiments. TD-NMR measurements were carried out on a Bruker Minispec mq20 equipment operating at a resonance frequency of 19.9 MHz for protons. After soluble extraction and prior to the NMR experiments, the networks were kept at room temperature for about 1 month to ensure a homogeneous distribution of guest chains through the network structure. Sample temperature was controlled with a Bruker BVT3000 unit with a stability accuracy of 0.1 K. Slices of 1 mm of sample were centered in 10 mm outer diameter standard NMR glass tube in order to maximize the homogeneity of the radio-frequency field. Transverse relaxation decay data were acquired using a compensated CPMG pulse sequence, with an MLEV-4 pulse phase cycling for the refocusing π pulses.22,23

EXPERIMENTAL METHODS

Materials. A set of model polymer networks were synthesized with different concentration of linear (guest) soluble polymer chains (see details in Table 1). Model PDMS networks were obtained by a hydrosylilation reaction between a commercial difunctional prepolymer, α,ω-divinylpoly(dimethylsiloxane) (B2: Mw = 21 300 g/mol; Mw/Mn = 2.95) (United Chemical Technology, Inc.) and a trifunctional A3 phenyltris(dimethylsiloxy)silane cross-linker. Networks were prepared by adding small concentrations of unreactive chains (B0) to the stoichiometrically balanced reacting mixture of B2 and A3.12,13 The unreactive linear chains were obtained by neutralizing the vinyl group chain end of monofunctional polymers by a hydrosilylation reaction with a monofunctional reagent, pentamethyldisiloxane. The molar mass characterization of the unreactive guest chains B0,i=1,2 are listed in Table 1. The system is then composed of a fully reacted A3 + B2 network, pendant material consisting of partially reacted groups that are linked to the network by one end and soluble material from unreacted or partially reacted precursors and B0 guest chains (see scheme in Figure 1). In order to reduce the complexity of the system, here we take advantage of the difference in molar mass between soluble molecules to remove the unreacted B2 solubles while keeping a small fraction of

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RESULTS AND DISCUSSION Swelling Experiments. The dependence of the extraction efficiency on the molar mass of the free chains is related to the number of physical entanglements in which these chains are involved. Long chains or chains with complex branched architectures will be subject to a greater amount of conformational rearrangements, decreasing their capacity to diffuse through the gel and thus increasing the extraction time. Different nonidealities during the reaction may lead to the appearance of molecules with complex architecture chemically unattached to the network. The scheme in Figure 2 shows examples of plausible linear (a−d), branched (e−h), and topologically trapped molecules (i−l) produced by the nonideal reaction of A3 and B2 molecules. As the complexity and molecular size of the soluble molecules increase, it can be expected that their extraction from the network becomes more difficult or even impossible in the presence of trapped entangled loops. Although these complex molecules can be present in the network, we note that in our case their content must be negligible since the maximum extent of reaction is quite large (see discussion below). Therefore, in the present study most of the soluble material is constituted by guest chains B0 and linear molecules formed by B2 precursors (top panels in Figure 2). For the systems studied here we have found that the volume fraction ϕ of polymer in networks isothermally swollen up to equilibrium with toluene reach values as low as ϕ ≲ 0.25. Under this degree of swelling B2 chains can be easily removed from the network due to their fast diffusional dynamics. In entangled polymer melts diluted by Θ-solvents the molar mass between entanglements Me(ϕ) depends on polymer concen-

Figure 1. Schematic representation of model PDMS networks obtained via the end-linking technique. Here the trifunctional crosslinkers are indicated with circles, elastic difunctional B2 chains with blue lines, and linear guest chains with yellow lines. An entangled loop that contributes to the network elasticity is indicated with a black line. Unreacted (green lines) or partially reacted (red lines) B2 chains led to soluble and pendant material, respectively. B


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Macromolecules

Figure 2. Different structures of defects that can be generated within the network due to the incomplete reaction of the precursors. Observe that the structure of these molecules can be linear (a−d) or branched (e−h) and that the material chemically unattached to the network can be also permanently trapped due to the formation of entangled loops, contributing to either the pendant or elastic structure (i−l). −4/3 14 tration ϕ as Me(ϕ) = Mmelt , where Mmelt is the molar e ϕ e ∼ 104 g/mol mass between entanglements in the melt (Mmelt e for PDMS). Thus, under equilibrium swelling conditions the effective molecular mass between entanglement for a trapped −4/3 ∼ 6Mmelt > MB2. linear polymer results Me(ϕ) ∼ Mmelt e 0.25 e Subsequently, in the swollen state most of the soluble chains in PDMS networks are completely unentangled and can be easily removed. On the other hand, B0 chains involve a much slower relaxation time and therefore require longer extraction times. Table 1 shows the fraction of extracted soluble material (Ws) that corresponds to B0 chains and unreacted B2 ones. It can be inferred from this that the nonreactive B0 guest polymer incorporated into the network during the reaction cannot be completely recovered and that the efficiency of the extraction of free chains decreases as its molar mass increases. For the guest chain molar mass and concentrations used in this work, the fraction of unextracted material ranges from ∼1 to 6%. Given that the contents of both guest and unreacted B2 chains are relatively small, it can be assumed that there are no self-entanglements among these defects. It can then be expected that the extraction rates of the populations of each species turn out to be independent of each other. Once the reaction and swelling experiments have been completed, the system is composed of fully reacted B2 chains that form the network, pendant material consisting of partially reacted B2 chains that are linked to the network by one end, and the remaining unextracted fraction of B0 guest chains (see Table 1). One of the simplest models to link the network architecture with equilibrium properties is the Miller−Macosko model that allows determining, for example, the fractions of elastic, dangling, and soluble material in terms of the extent of reaction p. This model also allows the correlation of network properties to viscoelastic response. For instance, according to the classical theory for rubber elasticity,13,15−20 the equilibrium properties depend on the concentration of elastically active chains and cross-linking points, while the dissipative dynamics depend on the concentration and architecture of the pendant

material, parameters that can be easily determined through the model. As in the network without linear guest chains (B2-00) there is a high efficiency in the elimination of solubles B2, the content of elastic and pendant material can be computed through the fraction of solubles and the Miller−Macosko model.20,21 According to this model, the maximum extent of reaction p can be obtained through the fraction of solubles Ws: Ws = waα3 + wbβ2, where wa and wb are the mass fractions of cross-linker and difunctional chains, respectively, and α = (1 − rp2)/rp2 and β = 1 + rp(α2 − 1). For this system, the stoichiometric imbalance r is computed as r = f [Af ]0/2[B2]0, where [Af ]0 and [B2]0 are the initial concentrations of cross-linker and difunctional chains, respectively (see Table 1). The fractions of elastically active We and pendant material Wp can be determined as We = wa((1 − α)3 + 2α(1 − α)2) + wbβ2 and Wp = wa(3α2(1 − α) + α(1 − α)2) + wb2β(1 − β). Taking into account the content of solubles for the network without linear guest chains (B2-00 in Table 1) results in p = 0.95, We ≃ 0.89, and Wp ≃ 0.11. It can be noted that the standard Miller−Macosko model does not account for the presence of loops, which affect the concentration of soluble material and network architecture. However, as in the present study the reaction is carried out without diluents and B2 precursors are large and flexible enough to have a Gaussian end-to-end distance, the concentration of loops can be expected to be negligible. We note that the maximum extent of reaction in the networks explored in this work is similar to the values of the literature,20 where it was also found a negligible incidence of loops upon an adequate selection of the molecular size of the precursors. Additionally, as 1 − p and 1 − rp are respectively the probabilities that an A or B group remains unreacted, and here the maximum extent of reaction reaches a relatively high value (rp ∼ p = 0.95), it can be concluded that the content of soluble material with complex architectures must be very small. For example, the content of linear soluble chains with about twice the molecular weight of the B2 precursors (see panel d in Figure C


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Macromolecules 2), formed by reaction of two of the three reactive groups of the A3 cross-linker and two B2 chains, is less than 0.01 wt %. As most of the soluble material with complex architecture has a relatively small molar mass and the selective extraction procedure is conducted under swelling conditions, it can be expected that most of these molecules leave the network during the extraction. Elastic Fraction Relaxation. In the analysis of the 1H NMR experiments, the transverse magnetization decay can be described as the sum of a solid-like contribution, with relative weight we, coming from elastically active chains and transient entanglements persisting on the NMR time scale, in addition to a liquid-like contribution, with relative weight wd, coming from the fraction of relaxed guest and pendant material.11 These values, we and wd, are extracted from the signal time-based decay by a nonlinear least-squares fitting procedure, as shown in the inset of Figure 3, where the fraction of elastic material is plotted as a function of temperature.

are the temperature-dependent fractions of unrelaxed pendant and soluble (linear guest) material, respectively. A decrease of the ∼1% of the elastic fraction we(T) is observed as the temperature increases (see Figure 3) for the network B2-B0,2-05, while for the network B2-B0,2-20 we changes from 87% to almost 84% for the same temperature range. This different behavior should be expected if we consider that networks B2-B0,2-05 and B2-B0,2-20 exhibit different fraction of guest chains (see Wsw g in Table 1). Previously, it has been found that during the process of network formation the maximum extent of reaction remains unaffected by the presence of the guest chains.24 This feature relies on two experimental observations: the dependence of the equilibrium modulus and the width of the relaxation spectrum with the content of linear guest chains. It was found that guest chains only have a “solvent-like” effect on the network elasticity. As in equilibrium guest chains cannot contribute to the network elasticity, the scaling of the equilibrium elastic modulus with the concentration of guest chains is consistent with the theory for semidilute Θ solutions. In addition, it was also found that upon appropriate rescaling the relaxation spectrum associated with the loss modulus G″(ω) for networks containing different contents of guest chains overlaps quite nicely over a wide range of frequencies ω.24 In addition, due to dilution, the effect of self-entanglements between guest chains is quite small since their distribution of relaxation times is roughly independent of the mass content of the unattached material. Based on these observations, it can be considered that the relative weight fraction of pendant and elastic material in the different networks is the same as the corresponding to the system without linear guest chains B2-00. The tube model theory combined with TTS can be employed to quantify the transient contributions of guest and pendant chains to we(T). The fraction of unrelaxed pendant gp(t) and guest chains gg(t) can be well described considering Rouse-like dynamics at short times scales combined with reptation dynamics25 for the guest chains and arm retraction for the pendant material.2 Note that within the network there are no dynamic dilution effects, and therefore the long time dynamics is dictated by “pure reptation” and arm retraction in a Pearson−Helfand potential.2 In order to describe the fraction of unrelaxed material within the network, it is important to consider the architecture of the different nonelastic constituents. We have observed above that the soluble material with complex architecture has a relatively small molecular weight and consequently is removed during the extraction. Therefore, unattached material is mainly formed by the unextracted fraction of the linear guest chains. Networks may also have material that is chemically unattached to the network but that is topologically trapped due to the formation of loops. However, under the experimental conditions of the present study it can be expected that trapped loops have a negligible contribution to the relaxational dynamics at the time scale of NMR experiments because (a) their content is well below 1 wt %, (b) low molar mass loops trapped by just one entanglement involve a relaxation time clearly below the relaxation time of the guest chains, and (c) loops trapped by several entanglements must behave as elastic material. Other important defects that deeply affect the dynamic response of polymer networks are pendant chains. Pendant chains are linked to the network structure only through one chain end, and their contribution has proven to be crucial in the description of the viscoleastic and dissipative long time

Figure 3. Elastic fraction as a function of the temperature for samples prepared with an initial content of 5% of B0,2 guest chains (circles) and 20% of B0,2 guest chains (diamonds). The inset shows a characteristic data set with the fitting functions described elsewhere.8

As the time scale of NMR observation is fixed around 1 ms, the transient solid-like contribution coming from the unrelaxed fraction of pendant and guest chains is affected by temperature. Upon temperature increase, a higher fraction of the chain ends of guest and pendant material loses its configurational memory and becomes isotropic at the time scale of NMR exploration. Hence, it should be expected that if the terminal relaxational time of defects becomes shorter than 1 ms, the solid-like contribution decays and eventually levels off at high temperatures. On the other hand, the fraction of elastically active chains and the permanently trapped entanglements generated by the cross-linkers cannot modify their contribution to the solid-like response with temperature because the elastic chains are linked to the network through both ends.20 The temperature-dependent fraction of elastic material we(T) determined by NMR can be expressed as we(T ) = We + g p(T )Wp + gg (T )Wg

(1)

where We, Wg, and Wp represent the fraction of elastic, guest, and pendant chains, respectively. Here gp(T)Wp and gg(T)Wg D


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Figure 4. Different structures of pendant defects that can be generated within the network due to the incomplete reaction of the precursors. Note that the structure of the pendant chains can be linear (a−d) or branched (e−g) and that these material can also contribute to the network elasticity due to the presence of trapped loops (h−j).

response. The scheme of Figure 4 shows representative pendant molecules that can be formed due to the uncomplete reaction of the network precursors. Previously we have shown that for a maximum of extent of reaction close to unity, more than 80% of the pendant material is linear and formed mainly by partially reacted linear B2 chains. We note that the pendant material with complex architecture may involve a very slow relaxation dynamics due to their branched nature. Therefore, at the NMR time scale of measurements this material behaves as elastic. Here we consider that the dissipative response of the network is exclusively dictated by linear pendants B2 chains and linear guest B0 chains. In order to determine the contribution of the linear guest chains and pendant material to the transient elasticity of the network, we consider that the dissipative dynamics can be described within the theoretical frame of the tube model. Since the slow dynamics of pendant and linear guest chains involve different mechanisms of relaxation, we employ two different approaches to obtain gp and gg in eq 1. Linear Guest Chains. According to the classical Doi− Edwards−de Gennes tube model, the relaxational dynamics of an entangled polymer chain can be separated into different regimes depending on the length and time scales. At very short times, the segments are considered to undergo unrestricted Rouse motion where the segments move freely without any constraint other than chain connectivity. At longer times, the chain “discovers” the constraints imposed by the confining tube, and then polymer conformation relaxes through a constrained-Rouse dynamics. At the Rouse time, the meansquare displacement along the tube is of the order of the chain end-to-end distance, retraction becomes increasingly slow, and most of the remaining chain segments relax their conformational memory by reptation. However, although this model

captures the dominant features of the entangled polymer dynamics, there are two important shortcomings:25 (a) the reptation model predicts that the reptation time τd scales with the chain size N as N3, while numerous experiments give τd ∼ N3.4, and (b) the tube model fails in the description of the polymer relaxation at intermediate time scales, below the longest Rouse time. For example, for frequencies above ∼τd−1 the Doi−Edwards theory predicts a dynamic loss modulus G″(ω) going as G″(ω) ∼ ω−1/2 while experimental data show a much weaker power law, with an exponent between 0 and 1/4 depending on chain length.25 In order to remove both discrepancies, Milner and McLeish considered a model that in addition to the standard Rouse and reptation dynamics includes a contribution from contour-length fluctuations;25 this contribution is determined considering the entangled linear chains as two-armed stars. At room temperature the terminal reptation time for the linear guest chains is τd ∼ 10−4 s and τd ∼ 10−3 s for B0,1 and B0,2 chains, respectively.24 From the terminal relaxation time τd and taking into account the contour length fluctuations, the Rouse time can be determined as τR = τd/[ne(1 − √ne)], resulting in τR ∼ 7 × 10−6 s and τR ∼ 7 × 10−5 s for B0,1 and B0,2 chains, respectively. At the millisecond time scale of NMR experiments the Rouse modes are completely relaxed, and their contribution is negligible. Then, in our case the dominant contributions to the anisotropic response become a sum of two terms: one corresponding to the loss of memory of the chain ends due to contour-length fluctuations and another one corresponding to escaping the remainder of the tube by reptation. Within the theoretical frame of Milner and McLeish, during the retraction the fraction of tube surviving from the initial configuration Φretr(t), can be expressed as25 E


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Macromolecules Φretr (t ) =

∫ exp[−t /τ(s)] ds

where L is the primitive path length, De is twice the Rouse diffusion coefficient, and U(s) is the arm retraction potential. While for star melts this potential must be modified to take into account the dynamic dilution,26 here the relaxation processes take place in a network where dynamic dilution is frustrated. (If the concentration of defects is small, ideally, below the entanglement concentration, the average tube diameter remains dynamically constant during the relaxation process.) As the relaxation proceeds in a permanent network, the potential has the following quadratic form:28,29 U(s) = 15/8nes2 (Pearson− Helfand potential). Previously, we clearly observed that this extremely slow relaxation mechanism in a Pearson−Helfand potential is responsible for the Thirion−Chasset power law behavior30 observed during the stress relaxation process of different polymer networks.31 Solving eq 7, we have2,24

(2)

where the retraction time τ(s) can be obtained as follows. At short times the chain moves under the action of many Rouse modes. In this case, this fast relaxation process is dominated by a one-dimensional Rouse-like dynamic characterized by τf(s), where τf (s) ∼

225π 3 4 4 τene s 256

(3)

Here τe is the Rouse time between entanglements, ne is the number of entanglements, and s (0 < s < 1) is the fractional distance back along the primitive path the free end have been retracted. This characteristic time dominates the dynamics at short times and is valid up to s = sd of order sd ∼ (1/ne)1/2. Beyond sd, retraction becomes increasingly slow, and most of the remaining chain segments will relax their configurational memory by reptation.25 We thus have 25

Φrept(t ) = Φretr (t = τs)

∑ p odd

8 exp[−p2 t /τd] π 2p2

⎡ 15ne ⎤ 1 τs(s , ne) = − Iπ 3ne 2τe erf⎢I s⎥ 2 8 ⎦ ⎣

where I = √−1 and erf[x] is the error function. In this case, the crossover between τf(s) and τs(s) can be also determined through the crossover formulas:26

(4)

τdangl(s , ne) =

As pointed out above, the Rouse modes that account for the stress relaxation on time scales shorter than τe and the longitudinal Rouse modes inside the tube that relax the configurational memory by redistributing chain segments inside the tube produce a negligible contribution at the NMR time scale of exploration. For example, it can be shown that for the longest guest chains B0,2 studied here, and for t = 1 ms, the Rouse contribution is ∼10−3 times smaller than retraction and reptation contributions. Then, the fraction of trapped entanglements gg(t) results: gg (t ) = Φretr (t ) + Φrept(t )

(5)

∫ exp[−t /τpend(s)] ds

(6)

where the retraction time τpend(s) for pendant chains is controlled by two different dynamic regimes. At short times, the dynamics is dictated by a one-dimensional Rouse-like relaxation, as described above by eq 3. Beyond sd, the remaining chain segments will relax their conformation by arm retraction. At long times (t ≳ τ(s = 1/ √ne), this slow relaxation process is controlled by the “first passage time” τs(s). The first passage time for this problem can be expressed as25,27 τs(s , ne) =

L2 De

∫0

s

s′

ds′ exp[U (s′)]

τf (s , ne)τs(s , ne) exp U (s) τs(s , ne) + τs(s , ne) exp U (s)

(9)

In order to determine the transient and permanent contributions to the network elasticity through eqs 1, 5, and 6, it is necessary to calculate the effective number of entanglements in which defects are involved. This average number of entanglements depends on both the molecular size of the defects and tube dimensions. Previously, we found that the relaxation spectrum measured through the loss modulus is insensitive to the content of guest chains, indicating that for the different networks studied here there are no important changes in the average number of entanglements involving defects and thus tube dimensions are not altered by the guest chains. In addition, on the basis of theory and rheological measurements, we have previously found that the tube dimension within the networks is equivalent to the tube dimension in well-entangled polymer melts.24 Irrespective of the polymer architecture (linear, star shaped, or networks), it has been found that above the melting temperature PDMS is thermorheologically simple and that time-frequency shift factors aT can be equally well described either by the Williams, Landel, and Ferry (WLF) model or the Arrhenius equation. 13 We can take advantage of the thermorheological simplicity of PDMS to obtain information in the time domain through data in the temperature domain. The time−temperature superposition principle was applied to we(T) data to obtain the fraction of elastic material as a function of time, we′(t). The corresponding horizontal shift factor, aT, was determined considering an activation energy Ea ∼ 29 kJ/mol13 and a reference temperature of T0 = 313 K. Motional regimes that NMR may detect are related to changes in the nuclear spin Hamiltonian, which in the present case is determined by the residual dipole−dipole couplings (Dres). Considering that for PDMS networks Dres is about 250 Hz,8 and for melts with chains of Mw around 130 kg/mol and less, Dres is ca. 100 Hz; the motional narrowing condition is in the range τ = 1/2πDres = 1.6−0.6 ms. Therefore, in this work we consider a characteristic time τ = 1 ms for the relaxation of the chains (see Figure 5). It is worthwhile to mention that within the temperature regime analyzed here the time window of

while the relaxed fraction [1 − gg(t)] behaves as isotropic. Dangling Chains. Differently from linear chains where the dynamic is controlled by the diffusion of the molecules along its own contour, pendant molecules cannot reptate to recover equilibrium. In this case, reptation is suppressed and the chains renew their configurations through arm retraction, a process in which the end of each arm independently retracts partway down its confining tube and then reemerges along a different trajectory.2,26,27 Following the theory for star melts, during the retraction of the pendant chain the fraction of the initial configuration surviving after time t, gp(t), can be expressed as g p (t ) =

(8)

∫−∞ ds″exp[−U(s″)] (7) F


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a similar relaxation spectrum that networks prepared with guest linear chains that present about 30 entanglements.24 On the other hand, it is worth to mention that due to the slow relaxational dynamics of the pendant material, the elastic fraction w′e(t) can reach an apparent plateau within the time scale of NMR exploration, which should relax beyond the terminal relaxation time of these defects. Elastic fraction we′(t) data for all samples as well as the fitting to eq 10 are shown in Figure 6. As the percentage of guest

Figure 5. Elastic fraction determined by NMR as a function of time (we′(t)) for network B2-B0,2-20. Insets: at the time scale of NMR testing, an unrelaxed fraction of guest gg(t) and pendant gp(t) material contributes to the solid-like response.

exploration increases by almost 2 orders or magnitude thanks to TTS (aT=253/aT=363 ∼ 70). Equation 1 is then expressed as we′(t ) = We + g p(t )Wp + gg (t )Wg

(10)

where We and Wg were determined by fitting NMR we′(t) data by eq 10 and are presented in Table 2.

Figure 6. Elastic fraction determined by NMR as a function of time (w′e(t)) for different networks. The solid lines are fits corresponding to eq 10.

Table 2. Experimentally Determined Parameters for the Fraction of Guest (WNMR ), Pendant (WNMR ), and Elastic g p NMR Chains (We ) Present in the Networks Determined by Fitting Eq 10 to the NMR Data network

Mn [kg/mol]

WNMR p [g/g]

WNMR e [g/g]

WNMR g [g/g]

Wsw g [g/g]

B2-00 B2-B0,1-20 B2-B0,2-05 B2-B0,2-10 B2-B0,2-20

0.0 47.8 97.2 97.2 97.2

0.110 0.110 0.108 0.107 0.103

0.890 0.886 0.871 0.861 0.833

0.00 0.004 0.021 0.032 0.064

0.01 0.02 0.04 0.06

chains present in the networks increase, the elastic fraction, We, decreases due to a larger contribution of partially relaxed defects, which act as a diluent to the network. An excellent agreement between the NMR and swelling results for the fraction of guest chains, Wg, within the network is obtained, as shown in Table 2. The temperature-dependent changes in B2B0,1-20 are significantly weaker than those obtained for the rest of the systems. This is due to the fact that the decay of gg(t) is faster due to the reduced chain length of B0,1 guest chains as compared to B0,2, rendering a time response that is driven mainly by gp(t)Wp + We.

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CONCLUSIONS In summary, we have shown that slow dynamics of PDMS network defects can be directly monitored by NMR spin relaxation of protons as a function of temperature. Application of the time−temperature superposition principle to transverse relaxation NMR parameters allows an accurate determination of the contribution of different type of defects to the relaxation processes. The temperature-dependent apparent defect contents, and their contribution to elasticity, can be well described by combining tube, reptation, and arm retraction concepts, at least in the time scale explored in our experiments. As hightemperature experiments ensure a complete relaxation of the guest linear chains, an accurate quantification of the content of elastic and pendant chains within the network is possible. Moreover, this result shows that time-domain NMR applied to dry polymer networks is a valuable alternative for structure determination and complements the indirect swelling and rheological methods traditionally used in polymer science.

In Figure 5, NMR data for network B2-B0,2-20 are shown with the corresponding fitting to eq 5; the contribution to elasticity of guest chains, gg(t)Wg, and that of the pendant chains, gp(t)Wp, are presented separately so as to be able to compare both contributions. It can be seen that the contribution of the free defects to elasticity is only relevant at short times. Note that although the number-average molar mass of guest chains is about 10 times larger than the one corresponding to pendant ones, both contributions run almost parallel to each other. This behavior is not unexpected since pendant material relax through the slow arm retraction mechanism. As pointed out above, even for moderately entangled systems the terminal relaxation time of pendant chains can be larger by several orders of magnitude than the corresponding to linear guest chains with similar molecular size. For instance, it was previously found through rheological experiments that pendant chains involving an average of about five entanglements present G


Article

Macromolecules

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(30) Chasset, R.; Thirion, P. Proceedings of the Conference on Physics of Non-Crystalline Solids; Prins, J. A., Ed.; North-Holland Publishing Co.: Amsterdam, 1965. (31) Vega, D. A.; Villar, M. A.; Alessandrini, J. L.; Vallés, E. M. Macromolecules 2001, 34, 4591−4596.

AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (G.A.M.). *E-mail [email protected] (D.A.V.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

The authors thank the Universidad Nacional del Sur, the Research Councils of Argentina: CONICET, ANPCyT PICT’s 2010-1096/2274 and SeCyT-UNC which supported this work.

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Contribution of Linear Guest and Structural Pendant Chains to

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