Branch Point Withdrawal in Elongational Startup Flow by Time

May 27, 2016 - We present a small angle neutron scattering (SANS) investigation of a blend composed of a dendritic polymer and a linear matrix with co...
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Branch Point Withdrawal in Elongational Startup Flow by TimeResolved Small Angle Neutron Scattering N. Ruocco,*,†,§ D. Auhl,⊥,# C. Bailly,⊥ P. Lindner,∥ W. Pyckhout-Hintzen,§ A. Wischnewski,§ L. G. Leal,† N. Hadjichristidis,¶ and D. Richter§ †

University of California Santa Barbara, California 93106-5080 Santa Barbara, United States Jülich Centre for Neutron Science-1, Forschungszentrum Jülich, D-52425 Jülich, Germany ⊥ Université Catholique de Louvain, B-1348 Louvain-La-Neuve, Belgium # Maastricht University, NL-6200 Maastricht, The Netherlands ∥ Institut Laue-Langevin, F-38042 Grenoble, France ¶ King Abdullah University of Science and Technology, 23955 Thuwal, Saudi Arabia §

ABSTRACT: We present a small angle neutron scattering (SANS) investigation of a blend composed of a dendritic polymer and a linear matrix with comparable viscosity in start-up of an elongational flow at Tg + 50. The two-generation dendritic polymer is diluted to 10% by weight in a matrix of a long well-entangled linear chains. Both components consist of mainly 1,4-cispolyisoprene but differ in isotopic composition. The resulting scattering contrast is sufficiently high to permit time-resolved measurements of the system structure factor during the start-up phase and to follow the retraction processes involving the inner sections of the branched polymer in the nonlinear deformation response. The outer branches and the linear matrix, on the contrary, are in the linear deformation regime. The linear matrix dominates the rheological signature of the blend and the influence of the branched component can barely be detected. However, the neutron scattering intensity is predominantly that of the (branched) minority component so that its dynamics is clearly evident. In the present paper, we use the neutron scattering data to validate the branch point withdrawal process, which could not be unambiguously discerned from rheological measurements in this blend. The maximal tube stretch that the inner branches experience, before the relaxed outer arm material is incorporated into the tube is determined. The in situ scattering experiments demonstrate for the first time the leveling-off of the strain as the result of branch point withdrawal and chain retraction directly on the molecular level. We conclude that branch point motion in the mixture of architecturally complex polymers occurs earlier than would be expected in a purely branched system, presumably due to the different topological environment that the linear matrix presents to the hierarchically deep-buried tube sections.

I. INTRODUCTION Studies of complex polymer architectures in external fields like shear or elongational flow which closely resemble processing conditions, have recently become possible by means of neutron scattering in well-controlled conditions. Small angle neutron scattering (SANS) and neutron spin echo (NSE)1−14 are complementary techniques that allow a microscopic focus both on the structure and on the dynamics, respectively. In the past, these methods have conclusively contributed to an enhanced understanding of the rubber elasticity of polymeric networks and melts in the quenched state. Especially the SANS technique, combined with mechanical devices (e.g., strain rig), has provided an ideal basis to investigate a wide range of length and time scales in extensional flow,5,10−12 which is the primary focus of this paper. This work is the continuation of our previous publication,9 which focused only on the microscopic relaxation processes. SANS measurements of the anisotropic correlation function S(q) of strained polymer samples as the material is stretched © XXXX American Chemical Society

due to an applied stress allow the transient chain conformation to be probed at the molecular level. However, in order to access those nonequilibrium structures at the molecular scale, the sample must be maintained at rather low temperatures. This is the basis of the experiments described in this paper. In principle, in situ anisotropic data along principal directions of the stress tensor can be collected if the stretching flow were steady. However, steady elongational flows are very difficult to achieve, mainly due to technical and practical reasons. Another possibility is the time-resolved SANS method. However, for adequately high real-time resolution, unacceptably long collection times are required, even at high flux reactors (e.g., ILL in Grenoble). Therefore, to the best of our knowledge, the in situ quenching (i.e., below the glass transition temperature, Tg) Rheo-SANS method utilized in our former work9 that Received: December 28, 2015 Revised: May 23, 2016

A

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Experimental data is collected in time windows and thus represents an average scattering signal from the polymer response. The main scattering parameters as well as the main characteristic times of the linear chain and outer relaxed arms in the long time limit are already known from the former scattering analysis.9 We shall see that this combination of now experimental data and previously measured polymer parameters will allow a model-free evaluation of the deformation of the inner section of the dendritic polymer and a validation of the presence of branch point retraction23,24 in a polymer blend.

decouples the experimental and microscopic time scales is the best current approach to connect macroscopic rheological properties with the structure of polymeric materials at the molecular level. Until now, however, only a few attempts have been made to develop molecularly based models for either linear or branched polymers in the nonlinear regime. In the past, quenched shear and extensional Rheo-SANS experiments as well as rheological characterization were performed on linear polystyrene (PS) high-Tg-systems, but were only supported via relatively simplistic modeling of the structure factor.5,10−12 The first attempts to connect detailed molecular models for high-strain rheology with Rheo-SANS experiments were achieved via studies of the Doi−Edwards process of fast chain retraction in linear polymers1 and work on model H-shaped monodisperse materials.2,3,13 In particular, a recent steady state SANS investigation of linear chain deformation, while passing through a flow cell with position-dependent measurements of the anisotropy, showed that the existing models of chain orientation and stretch could roughly describe the experimental data.14 However, despite some good success for modeling the behavior of entangled polymer melts under strong nonlinear flows relevant for processing,15−21 fully predictive molecular models still need further insights and developments, especially for branched polymers, to incorporate and evaluate such concepts as branch-point withdrawal. Here, we continue the investigation, begun in9 of a blend composed of a 10% two-generation dendritic polymer and a long linear matrix. The branched polymer can be described as a three-arm star, with each star branch being terminated by two additional branches. The microstructure and short time dynamics of the branched polymer in this blend were characterized in our previous study through an in situ quenched SANS study of a fast uniaxial step-strain followed by controlled stress relaxation.9 Similar to earlier studies,5,10−12 the strain rate was chosen such that the relaxation dynamics of the outer arms could be studied, with the linear matrix and the inner branches both held in a quenched or “frozen” state as their relaxation time scales are well separated from the time scales of the outer arms. From this prior experiment, the main parameters could be determined for both the branched and linear matrix chains in good agreement with theoretical rheology-based estimates. The dendritic polymer acted as a natural “probe” chain in the mixture and this behavior led to the confirmation of a tube diameter relaxation time as part of the interchain-tube-pressure model22 as well as the time scales for arm retraction dynamics and the reptational motions of the embedding matrix. Yet, the first appearance of dynamics that was an indication of branch point motion could already be detected in the time domain at the crossover between the end of the relaxation process for the outer arms and the beginning of relaxation of the inner arms.9 In the present work, we will entirely focus on the branch point motion and retraction process by performing a nonlinear stretching experiment in the start-up mode. The chosen range of strain rates comply with complete relaxation of the outer arms (ε̇·τD, outer arm ≪ 1), and relaxation of nonlinear stretch conditions for the linear matrix (ε̇·τR, lin ≪ 1), but still allow prominent nonlinear chain stretching for the inner star-like sections of the branched polymer. The detection of the anisotropic neutron scattering intensity is made in specific time frames. The scattering vector range during the Rheo-SANS investigation allows both microscopic chain deformation and tube model parameters to be independently determined.

II. EXPERIMENTAL SECTION a. Materials. All 1,4-polyisoprene-based polymers were synthesized by anionic polymerization high vacuum techniques using appropriate linking chemistry. The synthesis as well as the full characterization of all samples was explicitly detailed in former publications.9,25 Here, we only summarize the necessary molecular features. The molecular weight Mw of the linear matrix component was 810 kg/mol with a polydispersity index Mw/Mn = 1.09 and a number of entanglements in the melt state Zlin = 181. This linear polymer is a statistical H/D random copolymer (D-content ∼32%) to avoid unfavorable Flory− Huggins isotopic interactions in mixing high molecular weight polymers and different architectures. The SANS signal of the pure random copolymer did not show any new q-dependence from correlations in the structure. The dendritic polymer has two generations of branches with an overall Mw = 250 kg/mol with Mw/ Mn = 1.07 (hyper 19,47). The architecture is a three-arm star (i.e., inner branches) with each inner branch having two branches emanating from their ends. The label (hyper 19,47) denotes the mass of outer and inner arms in kg/mol, being respectively 18.5 and 46.6 kg/mol. The degree of entanglement of these branches in the melt state is Zouter = 4 and Zinner = 10, respectively. The mixture with ratio 90/10 linear/branched by weight was dissolved in toluene, precipitated in methanol in the presence of 0.1% of an antioxidant (BHT) and thoroughly dried under high vacuum until constant weight. b. Experimental Conditions. The linear rheology of the blend and all of its separate components was described and discussed in a previous publication to which we refer for details.9 Basic characteristic relaxation times were derived using the Likhtman−McLeish26 and branch-on-branch (BoB) models27−29 for the linear and dendritic polymers, respectively. The characteristic relaxation times of the linear matrix at −10 °C using the WLF shift factors from that study are τe,lin ∼ 0.00076 s, τR,lin ∼ 25 s, and τd,lin ∼ 13520 s. On the other hand, the longest relaxation time of the outer arms of the dendrimer is τarm(1) ∼ 55 s.9 Elongational start-up rheological experiments were performed in the present study by using the extensional viscosity fixture (EVF) for the ARES (Rheometrics Sci. Ltd., USA) rheometer, equipped with a 2KFRNT transducer. In Figure 1, we show the experimental elongational viscosity growth η+E(t) for the blend taken at −10 °C and the curve corresponding to the linear viscoelastic prediction (LVE).9 The shear and extensional viscosity measurements were connected using the Trouton ratio of 3. This linear viscoelastic prediction was obtained by first fitting the linear rheology storage and loss moduli from9 with a sum of Maxwell modes, from which the relaxation modulus G(t) was extracted (see eq 1). After this, the associated linear-viscoelastic viscosity (LVE) in the zero strain rate limit was computed as + ηE,LVE (t ) = 3

∫0

t

8

G(t ) dt = 3 ∑ giτi(1 − exp(− t /τi)) i=1

(1)

For the modeling, we discretized the integral into the smallest set of Maxwell elements. The relaxation times τi were spaced by one decade and spanned a range from 10−1 to 106 s and fixed. The prefactors gi varied thereby between 0.62 and 0.003 MPa and were refined. A very satisfactory agreement was achieved with 8 Maxwell modes. The strain B

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that was described in our previous investigations.1−3,9,32 In this system, the sample is uniaxially stretched in a dry nitrogen (N2) atmosphere at constant strain rate. The latter is determined from the final sample extension ratio and the stretching time to reach the assigned value. Rectangular samples of 1 mm thickness with initial length-to-width ratio of 6:1 to minimize biaxial contributions were loaded between the sawtoothed grips of the custom-made strain-rig device. Sizes and conditions were carefully controlled before and after the experiment to minimize the presence of degradation or inhomogeneity due to bubbles, uneven thickness or cracks, thus achieving a uniform deformation. The fixture was precooled at the target temperature for 30 min and samples at the same temperature were rapidly loaded within 30 s.9,21 The measurements on the blend were performed under a continuous nitrogen blanket with three slightly different strain rates (i.e., ε̇ = 0.00012, 0.00018, and 0.00021 s−1). The loading and temperature control procedures for the samples were identical for both rheological and SANS experiments albeit the ARES (rheology) and strain-rig (SANS) were used, respectively. Start-up experiments were undertaken for various final macroscopic strain rates, sample extension ratios Λ and stretching times, as specified in Table 1. The extension ratio Λ is calculated as the ratio of final (L)-to-initial (L0) sample length, L/L0. Scattering intensities were collected both at 4 and 5 m sample-todetector distances yielding a scattering vector range 2· 10−2 ≤ q ≤ 2 × 10−1 Å−1 and 1.5 × 10−2 ≤ q ≤ 1.3· 10−1 Å−1, respectively. The isotropic reference was only measured at 4 m. The magnitude of the scattering vector q is defined as q = 4π sin (θ/2)/λN, where θ is the scattering angle and λN the neutron wavelength. The 2D detector data (128 × 128) were corrected pixel-wise for scattering of the doublewalled evacuated quartz mantle, background and channel sensitivity. Furthermore, they were normalized to absolute scale [cm‑1] via the incoherent scattering level of a 1 mm H2O standard. In the anisotropic case, sectors with opening angles of 10° were selected along the directions parallel and perpendicular, respectively, to the principle deformation axes. Therewith, 1D anisotropic intensities were extracted from averaging over these narrow azimuthal angles. The detector settings, corresponding to the q-values listed above, provide a compromise between sufficient counting statistics (in the q-range of interest) and the chosen strain rate, which is determined by the microstructure of the dendrimer. The sample elongation was monitored by a digital video camera located perpendicular to the sample. As noted above, all SANS experiments were performed at −10 °C at three different but similar strain rates 0.00012, 0.00018, and 0.00021 s−1 (see Table 1). Therefore, the associated long stretching times allow the collection of scattering data in time-windows of the order of minutes, i.e., 10 min for experiment I (5 m sample-to-detector distance with a sample aperture ϕ 7 mm) and 20 min for experiments II and III (4 m sample-to-detector distance with a sample aperture ϕ 2 mm). Because of the smaller sample size in experiments II and III (i.e., both

Figure 1. Experimental elongational viscosity growth η+E(t) data and linear viscoelastic prediction η+E,LVE(t) (LVE) of the blend at −10 °C. The viscosity growth η+E(t) for the blend (experiment I) with ε̇ ∼ 0.00012 s−1 was measured at −10 °C using the same strain rate of the SANS experiment. The viscosity growth function in the linear viscoelastic regime was calculated from ref 9. rate condition of the start-up measurement in Figure 1 was chosen identical to the SANS experiment, which is described below. In Figure 1, the LVE prediction and elongational viscosity growth data are in very good agreement over the full-investigated time range. At short times the data follow almost a perfect t1 scaling, η+E(t) = 3G0Nt where the shear plateau modulus G0N can be rationalized considering the approximation η+E,LVE(t) = σ(t)/ε̇ ∼ (σ/ε)t = G0Nt where σ and ε are the stress and strain, respectively. The short time behavior was associated with G0N ∼ 0.387 ± 0.001 MPa, which is fully in line with literature values for the entanglement plateau modulus for 1,4-cispolyisoprene.30,31 Furthermore, in Figure 1, the elongational viscosity growth η+E(t), for the startup extensional flow for the blend is clearly dominated by the linear matrix, which is in linear rheological response as can be observed from its characteristic times. In addition, the η+E(t) data do not show any evidence of chain stretch for the innermost branch of the dendritic star. However, since the contribution of the linear matrix (random H/D copolymer) to the SANS data is expected negligible in the blend compared to the dendrimer, any stretching or deformation of chain segments even in the deepest hierarchical level of the dendrimer should be accessible in the SANS measurement. Therefore, in the remainder of this paper, we only focus on the scattering behavior. SANS measurements were performed at −10 °C for startup of uniaxial stretch at the D11 small angle neutron scattering instrument (ILL, Grenoble, France) using the same in situ stretching apparatus

Table 1. Summary of the Main Parameters at −10 °Ca label expt expt expt expt expt expt expt expt expt

I − t1 I − t2 I − t3 I − t4 I − t5 II − t6 III − t7 II − t8 III − t9

ε̇(−10 °C) [s−1] 1.2 1.2 1.2 1.2 1.2 2.1 1.8 2.1 1.8

× × × × × × × × ×

10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4

Λ [−] 1.1 1.3 1.5 1.7 2 2.1 2.4 2.7 3

± ± ± ± ± ± ± ± ±

tstretch [s]

0.05 0.1 0.1 0.1 0.1 0.15 0.15 0.15 0.15

794 2186 3379 4422 5776 6183 7296 8278 9154

± ± ± ± ± ± ± ± ±

380 642 556 490 416 596 521 463 417

λ∥ [−] − 1.3 1.52 1.66 1.8 1.85 1.96 2.09 2.29

± ± ± ± ± ± ± ±

2ϕ* [deg] − 120 126 129 132 133 135 137 140

0.04 0.04 0.05 0.05 0.05 0.06 0.07 0.07

The strain rate ε̇, the macroscopic extension ratio Λ (i.e. the ratio of final (L)-to-initial (L0) sample length, L/L0), the stretching time tstretch., the microscopic chain deformation λ∥, and isotropy angle, 2ϕ*. In experiment I the macroscopic strain, Λ = 1.1 was used as isotropic pattern. The reported value of 2ϕ* is therefore corrected by ∼6° as described in the text. On the other hand, experiments II and III were subtracted with the a

1

“real” isotropic pattern, i.e., Λ = 1. The values of 2ϕ* have uncertainties of ±1°. The value of λ∥ is equal to − 2 + C

1 4

+ tan 2 ϕ* .

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Macromolecules length and width), a beam aperture of 2 mm was adopted instead of 7 mm, as used for experiment I. However, all the experiments showed comparable counting statistics on the detector despite the different aperture and sample-to-detector distance. Unlike our former quenched SANS study,9 the neutron beam was maintained open throughout the entire measurement, while the ILL software LAMP provided a precise split of the data into the time-domains. The average macroscopic extension ratio of each time domain is defined as Λ(t) = L(t)/L0 with L(t) = L0eε̇t, where L(t) and L0 are the average length of the sample at time t and the initial length of the sample, respectively. The true Hencky strain ε is then defined as ln(Λ). In order to facilitate our discussion of branch point motions and retraction processes of the dendrimer, which is the purpose of this work, it would be advantageous to be able to estimate the characteristic relaxation time for stretching of the inner branches of the dendrimer. In the absence of the outer branches, the Rouse time for the inner branches (linear segment with Z = 10) would be far too short to allow for stretching by the flow at the strain-rates mentioned above. However, it is evident that the extra friction associated with the outer arms increases the effective Rouse time (i.e., the time scale for relaxation of stretch) for the inner branches to a value that is large enough to allow them to stretch. If we examine the linear viscoelastic data from our previous study,9 and compare the first minimum in G″ for the branched dendrimer (which we may very crudely associate with the stretch relaxation time) with the minimum for the linear matrix, we may surmise that the relaxation time for the inner branches is significantly longer than the Rouse time for the linear matrix. Indeed, in previous studies, an effective stretch relaxation time τstretch. that accounts for relaxed contributions of the outer hierarchy of branches has been estimated graphically, by identifying the strain rate at which there is a significant departure from the predicted linear response, showing nonlinearity.13,21 From inspection of the literature on systematic studies of the startup behavior of branched polymers at strain rates above the critical value (i.e., ε̇·τstretch > 1), a common observation is that the critical Hencky strain, εc, at which the first signs of nonlinearity appear, is εc ∼ 0.17−0.34. The fact that there is a range of values is solely due to uncertainties in reading the onset times from graphical data, but corresponds to an extension ratio Λ = eεc ∼ 1.2−1.4. Such a critical extension ratio is observed in all elongational viscosity data in the literature. This observation can be validated in detail through the sensitivity of neutron scattering experiments for the quantification of anisotropic parameters along the main axes of deformation. Furthermore, the critical strain is temperature-invariant. Focusing now on the actual Rheo-SANS conditions, using ε̇ = 0.00012−0.00021 s−1 and the critical strain, the onset of nonlinearity is therefore expected from εc /ε̇ in the time range of 1460−2900 s at −10 °C. We note that this empirical estimate is in good agreement with the theoretical stretch relaxation time, which can be calculated from molecular theories for the inner generation of such a dendrimer. This independently confirms the empirical approach. Following the works of McLeish, Lentzakis, Blackwell and others for combs and hyperbranched architectures,13,21,33−38 this characteristic time can be estimated for our system as

τstretch = 5Z innerqarmϕinnerτarm(1)

to provide an interpretation to cover all possible structural aspects for understanding the polymer chain dynamics at all accessible length scales. In this respect, the scattering model developed by D. J. Read,39,40 based on the Warner−Edwards (WE) tube theory for networks41 will be used, where the recipe for deriving the complex structure factor for a mixture of complex architectures is presented. This approach takes into account the coupling of different degrees of freedom, which are associated with the microscopic time scales. In the theory, two sets of variables are defined as either quenched (e.g., tube conformation and strain) or annealed (e.g., fraction of relaxed chain ends and the chain configuration within the tube). By using this concept, a random phase approximation-based (RPA) model can be constructed for any type of branched structure and their blends. Since the dendritic polymer is the dilute component, the full description can be thought of as due to the branched polymer as a probe chain and assuming that most of the scattering signal came from it. This is not unexpected in view of the concentration of the dendritic polymer. The volume fraction ϕhyper is of the order or slightly less than the overlap volume fraction ϕv* computed by using the span molecular weight of the dendrimer and taking into account the compactness of the star-like polymer through the so-called g-factor. In the present work, we maintain the same scattering procedure as in,9 except in this case, the anisotropic scattering is mainly due to the stretching of the inner arms and the outer arms contribute only as isotropic background. Since the scattering intensities were collected continuously during the start-up experiment, without the quenching step, the choice of the q-range is important. The standard or general procedure of collecting equilibrium or time-decoupled (i.e., quenched state) neutron scattering data at different sample-todetector distances and thereby enlarging and scanning a large qrange is not applicable here. Thus, we are limited to sample-todetector distances that represent the best compromise between good statistical data and the short time scales needed to cover the relevant length scales. Furthermore, the situation becomes worse as the deformation is increased. In particular, along the uniaxial direction the structure factor sensitivity is shifted toward even lower scattering vectors and relevant information at the chain deformation level quickly moves out of the detector window. On the other hand, the compressive direction remains accessible and moves toward higher q-values, but its sensitivity is much lower due to the inverse square root dependence on the deformation in an incompressible system. In our previous work, we were able to use different sample-todetector distances and thus had more data points toward low qvalues. There, we could calculate the ratio of the scattering intensities in the parallel and perpendicular directions over the full scattering vector range. This scattering ratio representation was extremely valuable due to the peaked appearance of the structure profile. Unfortunately, at present due to the compromise, the lowest q-part is missing. However, the evaluation or control of the molecular parameters over the full q-space can still be realized, due to a so-called iso-angle treatment, which is q-invariant and is described in the following section. This approach replaces the tedious sector averages along different orientation angles as shown for example by Abetz et al.42 Because of the complexity of the analysis involved in Read’s approach of the random phase approximation (RPA), we omit a detailed description of the derivation of the correlation functions and partial structure factors; instead we refer to the

(2)

With the number of arms qarm = 2, the number of entanglements Zinner = 10, the volume fraction ϕinner ∼ 0.5, and the longest relaxation time for outer arms τarm(1) ∼ 55 s, we obtain ∼2750 s at −10 °C for the stretch relaxation time. The time τarm(1) is estimated from the LVE data obtained in the previous study.9

III. THEORETICAL AND EXPERIMENTAL SCATTERING APPROACH In general, the low q-behavior of SANS experiments can be analyzed by using the simple Guinier or Zimm model to determine the basic parameters, e.g., the deformation of the full chain or parts of it and the isotropic state. However, our goal is D

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progressively release from their tube, thus the integral limits have to be adjusted as well as the cross-correlations evaluated between in-tube deformed and free isotropic segments. The actual structure factor in the dendrimer/linear blend has a similar form as the linear case, but is rather written in terms of an effective chain spanning two arms and modified for all other existing correlations between branches as required by the RPA model. The expression (4) is derived for a chain constrained in a network and thereby assumes that the tube (i.e., primitive chain) deforms affinely with the imposed deformation (3), whereas the deformation of the chain within the tube is reduced as described by the original Doi−Edwards theory. In the present work, we will use comparisons between our experimental scattering data and the theory to not only determine whether the actual chain deformation follows the theory, but also in case it does not, to calculate the best fit estimate of the actual (nonaffine) deformation of the primitive chain. The latter has been denoted in the previous discussion as λ. In the low q-limit, the argument of eq 4 leads to terms in Q∥2 λ∥2 + Q⊥2λ⊥2. Now, an effective deformation at the level of individual chains and subchains in the 2-dimensional detector plane can be postulated along any particular direction Q in terms of the components Rg,μ. Since q2 = q∥2 + q⊥2, where q∥ and q⊥ represent the q-values along the directions perpendicular and parallel to the principle axis of deformation, this leads to

original literature and our former publication for details regarding the present blend.9 The sole parameters of the full model are the microscopic extension ratio λ of the polymer chain, the fraction of relaxed chain ends of the second linear component fel, the tube diameter d and the nonlinear retraction coefficient γ (or equivalently α) that accounts for the nonlinear tube length equilibration. The network-based Warner−Edwards model accounts for different degrees of deformation depending on the length scale. For q→ 0, the chains deform affinely with the macroscopic sample extension ratio Λ, whereas for q-values larger than the inverse tube diameter, the local deformation is strongly under-affine. In the case of network-like systems, the absence of any possible retraction strengthens the assumption of affine motion at the level of the whole chain. In this context, the applied deformation tensor E in the principal axis coordinate system takes the form: ⎡ Λ1 0 0 ⎤ ⎢ ⎥ E = ⎢ 0 Λ2 0 ⎥ ⎢ ⎥ ⎣ 0 0 Λ3 ⎦

(3)

where Λ1 = Λ, Λ2 = Λ3 = 1/√Λ in which incompressibility is assumed and Λ is related to the true or so-called Hencky strain ε= ln Λ = ε̇·t. Considering that any distance ri̅ is transformed affinely as ri̅ → Eijr j, the WE structure factor for a tubeconstrained chain in a network under deformation (Λ = λ) is derived as S(q) =

∫0

1

dξ

∫0

1

⎧ ⎪ dξ′ exp⎨−∑ Q 2μΛ2μ|ξ − ξ′| ⎪ μ ⎩

+ Q 2μ(1 − Λ2μ)

λ 2(ϕ) = λ 2 cos2 ϕ + λ⊥2 sin 2 ϕ

The terms in this expression are the components of the chain deformation parallel and perpendicular to the principle uniaxial deformation axis. Here, ϕ is the angle between the scattering vector q and the principle direction of the applied uniaxial deformation. The parallel uniaxial and perpendicular compressive components of the actual deformation of the chain can be determined by identifying the so-called isotropy angle, ϕ*.46 The consequence of choosing this direction is that only isotropic scattering is observed, any contribution of the second term in eq 4 is effectively canceled out and is identical in this direction to the scattering from the undeformed sample. It thus follows that we can determine the scattering associated with the actual deformation of the chain by relating the data from the isotropic reference state to the anisotropic data. Since the tube diameter in the long time limit does not depend on deformation9 and thus can be considered constant, the isotropic angle represents an elegant and sufficiently accurate solution to directly extract information about the (nonaffine) deformation of the primitive chain.46−48 The determination of the isotropic angle proceeds by intersecting the isotropic and anisotropic scattering patterns. The intersection can be visualized by either subtracting or dividing the 2D-patterns, and either zero or unity-valued contour lines can be obtained, respectively (see Figure 2). In addition, the realized experimental deformation of the primitive chain, λ∥, can be independently accessed numerically by equating λ(ϕ) to 1.0 in eq 6. Then, a simple relation between the “special” angle ϕ* and the parallel deformation results:

d2 2 6 R g2

⎡ ⎛ 2 6 R 2|ξ − ξ′| ⎞⎤⎫ ⎪ g ⎢1 − exp⎜ ⎟⎥⎬ ⎜ ⎟ 2 ⎢⎣ d ⎝ ⎠⎥⎦⎪ ⎭

(4)

where the dimensionless contour length variable 0 < ξ,ξ′ < 1 describes the position along the chain. The components Qμ of the vector Q are related to the components qμ of the scattering vector q according to Q μ = R g, μqμ

(6)

(5)

where Rg,μ is the projection of the radius of gyration (of the deformed chain) in the direction μ. The WE structure factor contains the form factor of a deformed chain in terms of an affine contribution with the components of E (i.e. Λμ with μ = x, y, z), and a nonaffine contribution (i.e., second term of the argument) due to a reduction of stretch caused by fluctuations of the chain within the tube that is also a function of the tube diameter, d. In the original reptation literature,43,44 the parameter d represents the length scale associated with the confining potential for segment fluctuations around the primitive chain axis.43−48 For the tube diameter, d tending to 0, i.e., a strong confinement, the second term of the argument vanishes and the WE structure factor results in a purely affine deformation at the chain level. The result is an affinely transformed Debye scattering law. On the other hand, for d → ∞, the structure factor approaches the isotropic chain behavior and is devoid of any topological hindrance due to entanglement with other chains. For relaxing linear polymers, the chain ends

λ =− E

1 + 2

1 + tan 2 ϕ* 4

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In Figure 3, we present the theoretical computation of absolute scattering intensities in terms of λ∥, feo = 1 and d = 32

Figure 2. Representative iso-angle plots for different microscopic extension ratios of λ∥ = 1.52 for t3 (top left), λ∥ = 1.8 for t5 (top right), λ∥ = 1.96 for t7 (bottom left), and λ∥ = 2.29 for t9 (bottom right), obtained from a subtraction of anisotropic and isotropic scattering (see text). The red lines mark the angle and direction in which the scattering of the above 2D plots is identical to the isotropic data on the same sample. Since only zero-difference contour lines are shown, the data outside the red lines are due to the lower statistics at higher q. The stretching direction is vertical and the q-range shown extends up to q = 0.1A−1.

Figure 3. Representative anisotropic scattering intensities in the 2Ddetector plane of λ∥ = 1.52 for t3 (top left), λ∥ = 1.8 for t5 (top right), λ∥ = 1.96 for t7 (bottom left), and λ∥ = 2.29 for t9 (bottom right). The stretching direction is vertical. The q-range in each direction extends up to q = 0.1 A−1. Red lines are computations of the RPA structure factor with the deformation of the inner branches, obtained from the iso-angle and with a constant tube diameter from the relaxation study for t → ∞. The main values of the SANS calculations at −10 °C are d = 32 ± 2Å, fel = 0.25 ± 0.05, feo = 1 and λ∥ = var. (see Table 1). For the definition of these, we refer to ref 9.

Some examples of applying this procedure to our present data are shown in Figure 2 and the corresponding results are summarized in Table 1. This method crucially relies on wellcalibrated scattering intensities. Since the reference isotropic pattern for the smallest deformation states is lacking, instead of the isotropic reference with Λ = 1, we used the data for a deformation of Λ = 1.1. In that case, we corrected the soobtained isotropic angle due to the prestretch by the calculated offset of ∼2.95°, obtained from eq 7. This procedure is correct and valuable within the experimental uncertainties. Once the chain deformation of the inner arm section is quantified by means of the isotropic-angle construction, both 2D and 1D fitting of the scattering intensities can be confidently addressed using these values. The sensitivity of all the parameters involved in the scattering modeling was reported in our previous work9 and covered the full time range of the present study. That study dealt predominantly with the dynamics of the second (i.e., outer arm) generation of the dendritic polymer, whereas in the present study, no deformation of the outer arms is expected due to the low strain rates (i.e., slow deformation process). Therefore, basing our analysis entirely on the former study, we consider that the dangling end fraction of outer arms feo in Ruocco et al.9 has achieved its maximal value feo = 1. Although the outer arms are fully relaxed, they account for only 50% of the mass of the dendritic polymer, which is only 10% of the blend and lower or close to the overlap concentration. Their contribution to dynamic tube dilution (DTD)49,50 processes for the inner branches can be well neglected. Thus, the tube diameter that had stabilized at a value of ∼32 Å at the longest time9 can be kept constant in the present analysis. This last-mentioned parameter was estimated with very good accuracy from the intensity ratio representation as previously discussed.

Å. Very good-to-perfect agreement over the fully accessed experimental range in the 2D-detector plane is observed. In Figure 4, we report the same scattering curves via the ratio plot representation, S(q)⊥/S(q)∥. These 1D-computations are obtained by using the constant tube diameter, full dangling end contribution of the outer generation and the experimentally determined deformation of the inner branches of the polymer (see Table 1). The computations show that the nonaffine contribution enters in approximately at the length scale of ∼1/q = 1/0.02−50 Å, i.e., roughly twice the Warner−Edwards tube size.40 Furthermore, in our calculations the parameters of the linear matrix were chosen such as to match with the linear rheology expectations, although the linear matrix does not contribute to the scattering intensity visibly. The anisotropic behavior observed in the q-dependence is almost exclusively due to the branched (minority) component. It is well-known that dilute polymer systems make neutron scattering extremely valuable to obtain insight in multicomponent materials, whereas the macroscopic properties may be fully dominated by the majority phase. In the present context, the dendritic polymer acted as a suitable probe, whose dynamics is affected by the presence of the linear chain matrix material.9 The microscopic arm deformations, obtained through the isotropic angle procedure and summarized in Table 1, are displayed in Figure 5. There, the deformation of the primitive chain (i.e., corresponding to the inner branch of the dendritic polymer) is plotted versus the applied macroscopic extension ratio. Two distinct domains are observed in the data. For deformations up to roughly 1.5, i.e., corresponding to a critical Hencky strain εc = 0.4, the internal branches of the dendritic polymer deform approximately affinely with the applied F

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Figure 5. Microscopic chain deformation (λ) as a function of the macroscopic extension ratio (Λ). The solid red line is the theoretical deformation that only includes the retraction process without branch point effect (BP) and the dashed line represents the affine profile between microscopic and macroscopic strains. The stretch relaxation time from the retraction parameter α(t) is (4750 ± 1500) s and clearly underestimates the reduction of the microscopic deformation. Here, the value of t1, i.e., Λ = 1.1, is assumed as reference for the other deformations.

and α(t ) = 1 + [α0(Λ) − 1]e−t / τstretch.

At t = 0, the instantaneous retraction coefficient γ(t) is equal to 1, so no retraction appears, whereas for increasing times the parameter γ(t) approaches α0 (Λ). Here, α describes how the tube segments return to the state described by eqs 8 and 9. This treatment has been shown to be a good approximation in the evaluation of stretched H-polymers.3 The initial stretching of the inner chains is due to the coupling of deformation and orientation to an affine tube stretch factor α(Λ,t = 0). In the case of branched systems, as the retraction process involves the full inner arm, the stretch relaxation time τstretch. enters as the characteristic time to yield α(t) at any time, t. Hence, this retraction process was incorporated in eq 4 by replacing Λ2(i − j) in the first term with Λ2(i − j)/γ, which is essentially a rescaling process.40 It is a natural modification of the WE structure factor of a network toward the melt state. In the present context, the measured deformation is even lower than the remaining one after correction for the tube length equilibration process. We believe that this is a consequence of branch point withdrawal as will be explained below. Furthermore, additional insights can be observed by showing the time dependence of the primitive chain deformation (see Figure 6). Here, we show the measured deformation versus stretching time at fixed strain rate. Within our approximation, the modest variation of the strain rate within a factor of 2 does not lead to new phenomena or effects. The time scale of the start-up simulation experiment was normalized to the lowest strain rate of 0.00012 s−1. Data are essentially the same as in Figure 5 but seems to suggest more clearly experimental evidence that the inner branch of the dendritic chain is deformed differently in the two distinct domains. We note that the overall shape is not influenced by the merging of the strain rates and has the same trend observed in Figure 5. The crossover between these two

Figure 4. Ratio plot of the SANS data: top, from t1 to t5 (a); bottom, from t6 to t9 (b). The affine prediction with d = 0 is included as dashed lines. The main values of the SANS calculations at −10 °C are d = 32 ± 2 Å, fel = 0.25 ± 0.05, feo = 1, and λ∥ = var. (see Table 1).

macroscopic deformation. This effect may be due to the massive branch points that behave like cross-links inside a permanent network that typically follows the affine deformation assumption.51 The entanglements of the inner section may be treated as a dynamic network at short times. On the other hand, for deformations beyond 1.5, the experimentally measured chain deformation is found to deviate significantly from the applied deformation. In Figure 5, we show two curves: the straight line represents the pure affine deformation, i.e., λ∥= Λ and the second curve is calculated using the well-known Doi− Edwards result43 for the fractional reduction in the primitive chain length, α0 (Λ), after the tube segment stretch due to fluctuations within the tube: ⎡ ⎤ ln( Λ3 − 1 + Λ3/2) ⎥ Λ⎢ α0(Λ) = 1+ ⎥⎦ 2 ⎢⎣ Λ3(Λ3 − 1)

(8)

Similar to its use for former SANS studies on branched polymers,39,40 the retraction mechanism is theoretically simulated by considering the position of a monomer i to shift onto i/γ(t), where γ(t) =

α0(Λ) α(t )

(10)

(9) G

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system. Convective constraint release (CCR) is one such process. However, these mechanisms would lead to a reduction in the slope in Figure 6, rather than the transition that is observed. It seems clear to us that the most viable candidate to explain the data transition is branch point withdrawal. We admit that this is speculative, but we try below to explain how this might lead to the data shown in Figures 5 and 6. First, we note that branch point withdrawal (BPW) does not participate in the early and intermediate time range of the experiment (i.e., t < 1000 s). Since, the system is continuously deformed, the tension in the inner sections will progressively increase due to the constrained friction blob, until the point when the tension is sufficiently high to pull the branch point into the tube. We believe that the somewhat abrupt transition on the level of the arm deformation observed in Figures 5 and 6 is a consequence of the achievement of this point. The present experiment probes, in some respect, the force balance principle associated with branch point withdrawal (BPW) for the first time microscopically in a blend. Furthermore, Figure 6 allows us to estimate the stretch relaxation time for the supposed mechanisms. The overall time dependence of the initial part of the retraction process is observed to be in reasonably good agreement with (9) and (10) and provides the stretch relaxation time as τstretch. ∼ 4750 ± 1500 s at −10 °C. However, the retraction process does not lead to a reduction in the chain deformation, which is as large as the experimentally observed value. Indeed, the outer sections are included in the inner branches of the dendritic polymer leading to a modification in the measured deformation. As noted in the caption to Figure 6, the reduction in deformation is roughly equivalent to the lengthening that would occur if a single entanglement segment of the outer arms were pulled into the tube. The lengthening of the inner section corresponds also to a rise in the reference isotropic state chain dimension. Considering the resulting chain as a Gaussian coil with statistical segment length equal to the tube diameter d and a number of segments equal to the number of entanglements Z, the radius of gyration can be derived as Rg2 = Zd2/6. The incorporation of a single tube section, which belonged to two single outer relaxed arms now leads to an enhancement of this fictitious isotropic reference Rg by ∼1.05. This is equivalent to reducing the retracted microscopic deformation by ∼0.95, which is roughly 5% smaller. With this assumption, except for the transition regime the long time behavior of the chain deformation is thereby simulated in very good agreement and shown in Figure 6. Seemingly and surprisingly, branch point related processes show effects measurable by small angle neutron scattering techniques and these are obtained for the first time in an interesting mixture of different architectures. However, it is still to be understood why branch point withdrawal occurs when the macroscopic deformation Λ is still only about two, which violates the proposed priority model. For this concern, we propose the following interpretation, which is based on our former H-polymer investigations.3 To recapitulate, an Harchitecture is composed of four arms that are connected to a central backbone and can be considered as a two-arm comblike polymer. In order to retract two (or qarm) relaxed arms into the tube of this H-polymer backbone, the tube elongational factor α0(Λ) has to exceed ∼2 (or qarm) since for the priority pompom model α0(Λ) = ⟨qarm⟩, in order to allow branch point retraction.52,53 At α0(Λ) = 2, the equilibrium tension and the enhanced tension in the stretched backbone compensate. From

Figure 6. Dependency of the chain deformation (λ) as a function of the stretching time (tstretch.). The upper red lines (solid and dashed extension) represent a fit of the first few data points for primitive chain stretch, whereas the model does not include branch point effect (BP). The lower red line corresponds to the same theory but with the tube length increased by one entanglement segment as the result of branch point effect (BP) inside the tube. Here, the value of t1, i.e., Λ = 1.1, is assumed as reference for the other deformations.

domains occurs virtually in the same range of critical strain and extension ratio εc ∼ 0.4 respectively λ ∼ 1.5. The experimental time range that is associated with this critical deformation, shown in Figure 6, is found to be approximately 4500−6000 s. Figures 5 and 6 clearly show that the retraction of the inner arm, however, is insufficient to explain the experimental anisotropy of the SANS. Outer arm contributions can be excluded as the source, although this blob of relaxed material does contribute significantly to the drag required for retraction of the inner branch. We observe that the deviation increases with the deformation and that it seems to be correlated to the dynamics of the branch points. A discussion of the actual position in time and the level of deformation is therefore appropriate to better understand the branch point motion. Our observations suggest two main contributions to the time-dependent deviation from affine deformation. First, the effective microscopic deformation shows a time-dependence, which reminds one of the re-equilibration of the stretched tube contour length in the linear chain theory.43 We associate the second contribution with a nonlinear response of the branch point being withdrawn with the springlike chain, since the inner arms can be considered as stretchable sections between two branch points as in comb-like polymers. Parts of the outer arms are believed dragged into the tube of the inner sections of the dendritic polymer, whereas the motion of the friction blob limits the rate at which the inner chains can relax their elastic stretch. The static nature of the “cross-links”, both entanglements and branch points, enhances the tension in the chain. Therefore, at the intermediate time scale (i.e., the transition region), the deviation from affine deformation toward smaller microscopic deformations can be interpreted with the pure retraction process, which is described by eqs 8 and 9 without the incorporation of the branch point inside the tube. We note that in our experimental approach, the isotropic angle procedure automatically takes into account the reequilibration of the nonequilibrium tube contour length. Therefore, the model leading to (8) and (9) may lack additional relaxation processes that are present in the real H

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eq 8, this corresponds to λ∥ ∼ 4, rather than two as observed here. However, it should be noted that these ideas were developed for a pure one-component system whereas in the present case, we have a blend of branched and linear chains. Since this is a force argument and assuming that stresses are additive, we propose to introduce an average weighted force argument for a mixture of two components. By assuming the linear chain as a two-armed star polymer then qIarm = 1, whereas qIIarm = 2 is defined for the dendritic polymer. Reconstructing the force balance argument, we obtain that α0(Λ) > qIarmϕlin ν + qIIarmϕinner ν , which leads to α0(Λ) > 1.01. Since this is a very low value, we can use a Taylor expansion of α0(Λ) as 2 32 (Λ − 1)2 − (Λ − 1)3 5 105 5 76 + (Λ − 1)4 − (Λ − 1)5 + ... 21 385

REFERENCES

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α0(Λ) = 1 +

(11)

This corresponds to a minimum macroscopic deformation of Λ ∼ 1.2, which is in line with our estimation. However, a further investigation is planned in the future on this argument.

IV. CONCLUSIONS We have shown from time-resolved SANS experiments how in a bimodal architectural polymer mixture the basic mechanisms that are known to occur in conjunction with the specific architectural arrangements of the dendritic polymer remain active. We have shown here thathelped by the long relaxation times associated with the inner section of the branched polymerthe deformation behavior of the dendritic polymer could be accessed using time-resolved SANS in real-time. The scattering data obtained in the start-up phase complied with a network-like behavior from which the local deformation of chain sections between branch points could be determined in a model-free approach. We also showed a retraction of the inner arm of the dendrimer and finally signs of branch point incorporation in the scattering experiment. We thus observe branch point withdrawal for complex pompom architectures,52,53 though with some modifications required due to the blend with linear chains. Therefore, the present experiment proves that microscopic phenomena can be investigated through neutron scattering with good accuracy, provided the contrast for neutrons and therewith their visibility is welloptimized. However, in future investigations, both a detailed nonlinear rheological characterization and new deformation methods (e.g., shear experiments) will be necessary to provide additional insights into the topic.

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AUTHOR INFORMATION

Corresponding Author

*(N.R.) E-mail: [email protected]. Telephone: +1 (805) 893 8609. Fax: +1 (805) 893-5458. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors express thanks for the financial support of the EU (ITN DYNACOP 214627 Marie Curie Network). They also acknowledge Drs. Q. Huang, M. Shivokhin, and D. J. Read for helpful and productive discussions on the evaluation of nonlinear rheology and scattering modeling. We also would like to thank the Institut Laue−Langevin (ILL) for the availability of the neutron beam time. I

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