Relaxation processes in a lower disorder order transition diblock copolymer Alejandro Sanz, Tiberio A. Ezquerra, Rebeca Hernández, Michael Sprung, and Aurora Nogales Citation: The Journal of Chemical Physics 142, 064904 (2015); doi: 10.1063/1.4907722 View online: http://dx.doi.org/10.1063/1.4907722 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On the comparisons between dissipative particle dynamics simulations and self-consistent field calculations of diblock copolymer microphase separation J. Chem. Phys. 138, 194904 (2013); 10.1063/1.4804608 Effects of confinement on the order-disorder transition of diblock copolymer melts J. Chem. Phys. 124, 144902 (2006); 10.1063/1.2187492 Concentration fluctuation effects on the phase behavior of compressible diblock copolymers J. Chem. Phys. 120, 9831 (2004); 10.1063/1.1724819 Thermal composition fluctuations near the isotropic Lifshitz critical point in a ternary mixture of a homopolymer blend and diblock copolymer J. Chem. Phys. 112, 5454 (2000); 10.1063/1.481128 Ultra-small-angle x-ray scattering studies on order-disorder transition in diblock copolymers J. Chem. Phys. 110, 11076 (1999); 10.1063/1.479006
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THE JOURNAL OF CHEMICAL PHYSICS 142, 064904 (2015)
Relaxation processes in a lower disorder order transition diblock copolymer Alejandro Sanz,1 Tiberio A. Ezquerra,1 Rebeca Hernández,2 Michael Sprung,3 and Aurora Nogales1,a) 1
Instituto de Estructura de la Materia, IEM-CSIC. C/ Serrano 121, Madrid 28006, Spain Instituto de Ciencia y Tecnología de Polímeros, ICTP-CSIC. C/ Juan de la Cierva 3, Madrid 28006, Spain 3 Petra III at DESY, Notkestr. 85, 22607 Hamburg, Germany 2
(Received 28 July 2014; accepted 26 January 2015; published online 11 February 2015) The dynamics of lower disorder-order temperature diblock copolymer leading to phase separation has been observed by X ray photon correlation spectroscopy. Two different modes have been characterized. A non-diffusive mode appears at temperatures below the disorder to order transition, which can be associated to compositional fluctuations, that becomes slower as the interaction parameter increases, in a similar way to the one observed for diblock copolymers exhibiting phase separation upon cooling. At temperatures above the disorder to order transition TODT, the dynamics becomes diffusive, indicating that after phase separation in Lower Disorder-Order Transition (LDOT) diblock copolymers, the diffusion of chain segments across the interface is the governing dynamics. As the segregation is stronger, the diffusive process becomes slower. Both observed modes have been predicted by the theory describing upper order-disorder transition systems, assuming incompressibility. However, the present results indicate that the existence of these two modes is more universal as they are present also in compressible diblock copolymers exhibiting a lower disorder-order transition. No such a theory describing the dynamics in LDOT block copolymers is available, and these experimental results may offer some hints to understanding the dynamics in these systems. The dynamics has also been studied in the ordered state, and for the present system, the non-diffusive mode disappears and only a diffusive mode is observed. This mode is related to the transport of segment in the interphase, due to the weak segregation on this system. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4907722] I. INTRODUCTION
Block copolymers consist of at least two different chemical species that are covalently linked together forming a single macromolecule. Depending on the interaction of the molecular constituent blocks, they may tend to phase separate. However, this phase separation is tempered by the covalent bond that links them. This balance gives rise to a rich variety of ordered phases.1 A particular class of block copolymers, diblock copolymers, is formed by two segments of different polymers A and B with a single joint per chain. Most diblock copolymers exhibit a disorder to order transition upon cooling, and this is normally referred to as Upper Disorder-Order (UDOT). The reason for this phase separation upon cooling results from an increase of the interaction energy between the two chemically distinct segments, quantified by χ, the Flory-Huggins segment-segment interaction parameter. However, there are diblock copolymers that undergo microphase separation upon heating.2 The transition from disorder to order observed upon heating is normally named as Lower Disorder-Order Transition (LDOT). It has been shown that, whereas the UDOT is enthalpically driven, the LDOT may arise from differences in volume expansion between the constituent species at a)Author to whom correspondence should be addressed. Electronic mail:
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elevated temperatures, and therefore, it is entropic in its nature. This entropically driven ordering transition is accompanied by a positive change in volume upon demixing of the copolymer blocks.3 Due to this, phase separation can be avoided by pressurization,4 and these polymers are sometimes termed baroplastics. The balance between the entropic and enthalpic behavior has been postulated to be responsible for the peculiar phase separation of these systems5 that in some cases gives rise to a loop behavior where one can find a temperature window of miscibility, and above and below this window, the system is in two phases. Similar behavior has been found in particular blends, where the difference in compressibilities produces a phase separation upon heating.6 Extensive theoretical work has incorporated finite compressibility into the Random Phase Approximation (RPA) through effective interactions.7,8 In that way, χ was reinterpreted as χ = χapp + χcomp, where χapp is the conventional exchange energy with proper density dependence and χcomp represents compressibility difference between constituent blocks. Block copolymers with large compressibility difference and, thus, large χcomp exhibit a gain of entropy due to a volume increase upon phase separation. In any two polymer components A and B, either blends A/B or A-B block copolymers, composition fluctuations are behind the transition from the homogeneous state to the phase separated one, and this is evidenced by a maximum in the static structure factor S(q) at a finite value of the wave vector modulus q∗.9
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From the dynamics point of view, mean field theory for incompressible monodisperse systems10,11 predicts that these composition fluctuations relax via an internal chain mode or breathing mode with a relaxation rate that is independent of q. This mode arises from the relative translational motion of the center of mass of the blocks. For systems with heterogeneity in the composition along the block copolymer chains, Semenov et al. extended the formalism12 and predicted additional composition fluctuations in the form of another mode with a diffusive character, i.e., having a q2 dependence of its relaxation rate. This mode has been referred as the heterogeneity mode, and it is related to the self-diffusion coefficient of the block copolymer. Experimentally, techniques such as light photon correlation spectroscopy (LPCS) allow measuring directly the dynamic structure factor, by using time correlation functions. The scattered intensity correlation function, g2(q, t), can be obtained as ⟨I(q, 0)I(q,t)⟩ ⟨I(q,t)⟩2
.
(1)
In the decade of the 1990s, intensive work proved the existence of the internal mode13,14 and the heterogeneity mode12,15 in UDOT block copolymer melts and solutions16,17 mainly by LPCS. This technique, however, addresses the dynamic structure factor at length scales of the order of 500 nm.18 The use of highly brilliant and partly coherent X-rays, made possible in third generation synchrotrons, allows to use X-rays as a probe to observe the microscopic dynamics at the nanoscale by X ray photon correlation spectroscopy (XPCS). Because the X-ray wavelengths are small compared to those of visible light, XPCS is sensitive to the dynamics at much smaller scales. Recently, XPCS has allowed direct measurements of antiferromagnetic domain fluctuations in ferromagnets,19 slow dynamics in aerogel due to pore breathing,20 or the fluctuations originated in an crystallizing system.21 Related to concentration fluctuation in block copolymers, a few articles have been published in the last years where the dynamics of UDOT diblock copolymers in the disorder state has been studied,22–24 as well as quenched into the ordered state. The direct relation between S(q,t) and the composition fluctuations derived from the mean field theory12 is not valid for systems with finite compressibility, and therefore, the predictions for the existence of the internal mode and the heterogeneity mode prior and during phase transition do not necessarily apply. In this work, we present the first experimental evidence of the existence of two similar modes to those predicted by the mean field theory.9,11–13 By applying XPCS on a LDOT block copolymer, we observe two modes, a q independent internal mode and another with a q2 of the relaxation rate, similar to the ones predicted and observed for incompressible UDOT systems, but presenting some peculiarities by the fact that the system is a LDOT.
Inc.). Films of the sample were prepared by solving in toluene the as received powder and casted into glass substrates. In order to remove the solvent, samples were evacuated at 320 K for two days. Small Angle X Ray Scattering (SAXS) measurements were performed either by a Bruker Nanostar SAXS instrument or at the beamline Spanish CRG BM16 at the European Synchrotron Radiation Facility (Grenoble, France). The lab based Nanostar uses Cu Kα radiation (1.54 Å) produced in a sealed tube. The scattered intensity was corrected with the transmission of the samples calculated considering the absorption of the sample and that of the aluminum encapsulation. A glassy carbon standard was used for this purpose. In the case of measurements at the BM16, a complete description of the beamline setup can be found elsewhere.25 SAXS measurements were made placing the samples either between aluminum or Kapton® foils. The wavelength used in this case was 1 Å. All the SAXS patterns were normalized by the beam intensity, and an empty cell measurement was subtracted. The 2D isotropic SAXS patterns were azimuthally integrated to derive the SAXS curves as a function of the scattering vector q, q = 4π/λ sin(θ),
(2)
where λ is the wavelength and 2θ is the scattering angle. XPCS experiments were performed isothermally at different temperatures. At each temperature, series of above 400 SAXS patterns were collected, maintaining the total exposition time below 25 s to ensure that the sample was not damaged by radiation. Different spots on the sample were illuminated at every temperature. The experiments were carried out at P10 coherence beamline, PETRA III (DESY, Hamburg, Germany). The sample was placed under vacuum. The scattered intensity is recorded on a 2D Maxipix detector (developed at the European Synchrotron Radiation Facility, ESRF, Grenoble, France) with a pixel size of 55 × 55 µm2, 516 × 516 pixel active area, 13 bit/pixel dynamic range, and readout time of less than 0.5 ms. The experiments were performed at 7 keV and the sample to detector distance was 5 m. For each image sequence, the normalized intensity time autocorrelation function g2(t) was calculated pixel by pixel and finally averaged over pixels within a given range of q vector values using the XPCSGUI software.26 Oscillatory shear measurements were performed in a TA Instruments AR1000 rheometer, using samples obtained from toluene casted films. A 25 mm parallel-plate geometry was employed and the gap size was maintained at 0.3 mm. Isothermal frequency sweeps over four decades of frequency (0.01 Hz-100 Hz) were carried out at increasing temperature starting from 373 K to 463 K. A torque value within the linear viscoelastic region (torque = 100 µN m) was applied for all the measurements. III. RESULTS AND DISCUSSION
II. EXPERIMENTAL
The LDOT system studied here is a diblock copolymer of Polystyrene (PS) and Poly(Ethyl Methacrylate), PS-b-PEMA (Mn = 119k, PDI = 1.14, φPEMA = 0.55, Polymer Source,
The evolution with temperature of the SAXS intensity from initially disordered PS-b-PEMA is shown in Figure 1. At low temperatures, the system exhibits a broad peak in SAXS centered around values q∗ = 2π/d, where d = 1.86Rg, being
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FIG. 1. SAXS intensity during heating the initially disordered (not phase separated sample) from T = 390 K to T = 460 K. Measurements taken at BM16, ESRF.25 The inset shows the inverse of the intensity at the maximum (open circles) and full width at half maximum (filled squares) as a function of the inverse temperature during heating. Dotted lines indicate the TODTTDOT.
Rg the radius of gyration of the polymer chain. As temperature increases above 430 K, this peak becomes narrower and more intense, and its maximum shifts towards lower q values. The evolution of the peak characteristics, reciprocal intensity at the position of the maximum, I(q∗), and full width at half maximum (FWHM) as a function of the inverse temperature is presented in the inset of Figure 1. These results suggest that microphase separation is taking place, and from Figure 1, one can estimate the temperature (TDOT) at which the disorder-order transition occurs. For this particular system, we have previously reported a TODT TDOT = 430 K.27 The phase transition can be also corroborated by the development of higher order peaks in the static structure factor, particularly, a second and third order of the main peak, that indicates the formation of a lamellar structure can be observed, Figure 2. The temperature dependence of the interaction parameter for this system with slightly lower molecular weight was reported in the past by Ryu et al.5 The authors showed that χ for this system increases with temperature. In order to understand the dynamical origin of the phase separation observed by SAXS upon heating, XPCS experiments were performed isothermally at different temperatures, from 410 K to 450 K. The obtained autocorrelation function presents, at each temperature, a decay shape that can be modelled with Eq. (3), g2 (q, ∆t) = A + β · exp (−2[t/τ]α ) ,
FIG. 2. SAXS pattern from the “as cast” sample at room temperature (black) and at T = 450 K.
In the studied temperature range, upon heating from the homogeneous state, g2(q, ∆t) reveals a relaxation, with characteristic times in the range from 0.1 to 1 s (open symbols in Figure 3, top). In the bottom of Figure 3, the relaxation time τ, obtained from fitting of Eq. (3) to the relaxation curves, is presented as a function of temperature. Upon heating up to temperatures close to the TDOT, the observed relaxation becomes slower as temperature rises, as revealed by the observed increase in the relaxation time τ. However, as the TDOT is surpassed, the relaxation tends to speed up as temperature increases. These
(3)
where 1/τ and α are the relaxation rate and an empirical exponent that accounts for deviations from a simple exponential. When the value of α is between 1 and 2, the decay function takes the form of a compressed exponential, whereas for values between 0 and 1, the decay function is a stretched exponential. The factor β is a contrast factor that depends upon the beamline conditions and the detector used, and A is a fitting factor very close to unity. The value of A represents the contrast of the speckles, which is mainly determined by the coherent length of the X-ray and the pixel size of the detector.28,29
FIG. 3. Intensity-intensity autocorrelation function g 2, as a function of the elapsed time, obtained at q ∗ value (position of the maximum in SAXS) at different temperatures, upon heating from the homogeneous state (open symbols), and during subsequent cooling (filled symbols) ( ⃝) 410 K, (△) 420 K, () 430 K, (♦) 440 K, (✩) 450 K. In the bottom panel, the characteristic relaxation time is presented as a function of temperature.
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FIG. 4. Intensity-intensity autocorrelation function g2, as a function of the elapsed time, obtained at different q values upon heating from the homogeneous state, at temperatures below DOT (a, T = 410 K) and above DOT (b, T = 430 K). ( ⃝) q = 0.014 Å−1, (•) q = 0.015 Å−1, (△) q = 0.016 Å−1, and (N) q = 0.018 Å−1. Continuous lines in the both panels correspond to best fit to Eq. (3).
results indicate that there might be a change in the nature of the dominant relaxation in the proximity of the disorderorder transition. In fact, the relaxation exhibits a different q behavior in the two different temperature ranges. Figure 4 shows the correlation function for different q values at two selected temperatures, below and above TDOT . Whereas for temperatures T < TDOT, Figure 4(a), g2(t) is independent of the scattering vector and all the curves collapse into a single one, at temperatures T > TDOT, Figure 4(b), the characteristic relaxation time is smaller for higher q values. This can be easily observed by plotting the values of the relaxation rate 1/τ as a function of q2 for different temperatures, Figure 5(a). Below TDOT , in the disordered state, the relaxation rate does not depend on q2, indicating a nondiffusive character of the observed relaxation. This feature is what is predicted for the internal mode of the mean field theory for incompressible diblock copolymers.12 The peculiarity observed in the present case, is that, contrary to what is observed in block copolymers with UDOT,22,23 as temperature increases the relaxation rate decreases. This slowing down is similar to the one reported for UDOT systems when quenched below the TDOT and allows to transit from disorder to order.22 It is worth mentioning that the exponent parameter α in Eq. (3) takes values above 1 in this temperature range. This compressed exponential behavior has been observed for a variety of soft matter systems undergoing jamming transitions, which results in arrested solid like collective dynamics.30–32 As the temperature surpasses the estimated temperature of the transition, the relaxation becomes diffusive with a clear q2 dependence of 1/τ. The predominant mode in this regime, therefore, has the features of that predicted for incompressible
J. Chem. Phys. 142, 064904 (2015)
FIG. 5. Relaxation rate 1/τ as a function of q 2. Continuous lines are linear fits. (a) Heating scan. (b) Cooling scan.
block copolymers with some polydispersity, named in the literature as polydispersity or heterogeneity mode12 and might be related to interdiffusion processes at the interphases.33 The fact that the diffusive motion does not appear in the disordered region, below the TDOT , it might be explained by considering that, in this region, the dynamics is governed by concentration fluctuations that become more intense as the TDOT is approached. The compositional profile may have a weakly sinusoidal shape that becomes more intense as the TDOT is approached. As TDOT is crossed and the system enters a stronger segregation region, these fluctuations disappear and the composition profile evolves from a weakly sinusoidal profile, to a step function, with decreasing width of the interphase. In this situation, the observed relaxation corresponds to the diffusion of chains across the interphase leading to order improvement. LDOT block copolymers allow checking about the persistence of either of the modes in the ordered phase in a wide range of temperatures. This is so because, upon cooling from the ordered state to T < TDOT , the system thermodynamically should be disordered or mixed. However, the ordering prevails as has been observed recently by SAXS27 due to the dynamical arresting produced by the slowing down of the dynamics as the Tg is approached. In our case, the highest Tg block is the polystyrene component, with a Tg ∼ 363 K. The relaxation revealed by the autocorrelation XPCS signal during cooling is presented as filled symbols in Figure 3. The results show that, upon cooling, there is a relaxation process, even at temperatures below the TDOT , with no significant singularity when crossing the transition temperature. Analysis of the q dependence of this process on cooling indicates that it is diffusive along the whole range of temperature studied, exhibiting a q2 dependence of 1/τ (see Figure 5(b)). From the slope of 1/τ vs q2, the chain self-diffusion coefficient (Ds , 1/τ = D s q2) can be obtained. Figure 6
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J. Chem. Phys. 142, 064904 (2015) TABLE I. Relaxation time obtained from XPCS at q = qmax and terminal relaxation time estimation from rheology. T (K) 430 440 450
FIG. 6. Chain self-diffusion coefficient as a function of the inverse temperature of the heterogeneity mode during heating ( ⃝) and cooling (N) scans. The red dotted line indicates the temperature of the disorder-order transition, and the label 600 s indicates the annealing time at T = 450 K.
presents the dependence of D s as a function of the inverse temperature for the diffusive process in both the heating and the cooling scans. During heating, where, according to previous temperature resolved SAXS experiments27 at temperatures above TDOT , the system undergoes progressively the ordering or phase separation process, Ds is higher than the value obtained for the same temperatures during the cooling scan. These results indicate that the better is the order of the structure the slower the diffusion becomes. During the cooling scan, the sample has a quenched lamellar structure and, as revealed by the SAXS patterns, is isotropic. Because of that the obtained diffusion coefficient shown in Figure 6 is isotropically averaged. D s in the lamellar morphology exhibits a curved dependence that resembles a Vogel Fulcher Tamann dependency. When the lamellar structure is formed, the joints between the two different species are likely to be located at the interphase between lamella.
τXPCS (s)*
τtr (s)
0.84 0.67 0.60
18.1 13.2 13.1
Diffusion of a chain implies dragging one block into a region rich on the other component, and therefore, it will experience a thermodynamical penalty. This might be the reason for the retardation compared to the disordered or mixed state. These results are in accordance with recent Monte Carlo simulations based on both the slithering-snake dynamics and the slip-link dynamics,34 where it has been shown that self-diffusion is coupled in the parallel and perpendicular directions, and they predicted a significant slowing down of the dynamics in the lamellar phase compare to the homogeneous melt. Rheology experiments presented in Figure 7 of the sample at selected temperatures after time-temperature-superposition (tT) of the high-ω data show that there is a clear change in the behavior below and above the TDOT . At low temperatures, the experimental conditions do not allow to reach the terminal regime due to the proximity of the glass transition of the blocks. However, as temperature is increased, the rheology of the system start changing and time-temperature superposition does not apply. In the temperature range 430 < T(K) < 450, the system evolves, which can be attributed to the occurrence of the disorder to order transition. At high temperatures, a low-frequency power-law behaviour is found for both G′ and G′′ so that G′ ∼ G′′∼ ω0.5 which is characteristic of ordered lamellar block copolymers.4,35 Thus, confirming the ordered state found by SAXS experiments. At T > TDOT, an estimation for the relaxation time can be obtained from rheology experiments by τtr = G′/(ωG′′)]|ω→ 0.35 The values of the relaxation time, at T > TDOT during the heating scan, obtained from rheology and from XPCS are presented in Table I. Although following the same trend, the values for the terminal relaxation are 1.5 order of magnitude higher than the obtained from XPCS.
IV. CONCLUSIONS
FIG. 7. (a) Storage and (b) loss moduli as a function of the reduced frequency, aTω, for the initially disorder diblock copolymer. ( ⃝) 410 K, (△) 420 K, () 430 K, (♦) 440 K, (✩) 450 K.
We have studied the development of concentration fluctuations in a disordered LDOT diblock copolymer and the diffusive dynamics in the ordered lamellar system by XPCS. Upon heating, before phase separation occurs, a process with the characteristics of the internal mode predicted for incompressible systems is observed, besides the fact that clearly LDOT systems are not incompressible. This mode is peculiar in the sense of exhibiting a slowing down as T increases, which can be understood simply by considering the dependence of the interaction parameter χ with temperature: χ increases with increasing T, due to a competition between the two components contributing to it, an entropic and an enthalpic contribution, and upon heating, the entropic contribution
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becomes more important. Once the TDOT is surpassed, only a diffusion mode is observed. It is worth pointing out again that this diffusive motion becomes slower, but still present, when the lamellar structure is better defined. In this work, we have probed experimentally, for the first time to our knowledge, that the modes predicted for incompressible systems appear also in systems whose phase behavior cannot be explained without assuming compressibility. ACKNOWLEDGMENTS
Financial support from MINECO (Grant No. MAT201123455) is gratefully acknowledged. The experiments performed at P10 in PETRA III (DESY, Germany) were supported by the European Union (Nos. I-20110191 EC and I-20120060 EC). 1M.
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THE JOURNAL OF CHEMICAL PHYSICS 142, 064904 (2015)
Relaxation processes in a lower disorder order transition diblock copolymer Alejandro Sanz,1 Tiberio A. Ezquerra,1 Rebeca Hernández,2 Michael Sprung,3 and Aurora Nogales1,a) 1
Instituto de Estructura de la Materia, IEM-CSIC. C/ Serrano 121, Madrid 28006, Spain Instituto de Ciencia y Tecnología de Polímeros, ICTP-CSIC. C/ Juan de la Cierva 3, Madrid 28006, Spain 3 Petra III at DESY, Notkestr. 85, 22607 Hamburg, Germany 2
(Received 28 July 2014; accepted 26 January 2015; published online 11 February 2015) The dynamics of lower disorder-order temperature diblock copolymer leading to phase separation has been observed by X ray photon correlation spectroscopy. Two different modes have been characterized. A non-diffusive mode appears at temperatures below the disorder to order transition, which can be associated to compositional fluctuations, that becomes slower as the interaction parameter increases, in a similar way to the one observed for diblock copolymers exhibiting phase separation upon cooling. At temperatures above the disorder to order transition TODT, the dynamics becomes diffusive, indicating that after phase separation in Lower Disorder-Order Transition (LDOT) diblock copolymers, the diffusion of chain segments across the interface is the governing dynamics. As the segregation is stronger, the diffusive process becomes slower. Both observed modes have been predicted by the theory describing upper order-disorder transition systems, assuming incompressibility. However, the present results indicate that the existence of these two modes is more universal as they are present also in compressible diblock copolymers exhibiting a lower disorder-order transition. No such a theory describing the dynamics in LDOT block copolymers is available, and these experimental results may offer some hints to understanding the dynamics in these systems. The dynamics has also been studied in the ordered state, and for the present system, the non-diffusive mode disappears and only a diffusive mode is observed. This mode is related to the transport of segment in the interphase, due to the weak segregation on this system. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4907722] I. INTRODUCTION
Block copolymers consist of at least two different chemical species that are covalently linked together forming a single macromolecule. Depending on the interaction of the molecular constituent blocks, they may tend to phase separate. However, this phase separation is tempered by the covalent bond that links them. This balance gives rise to a rich variety of ordered phases.1 A particular class of block copolymers, diblock copolymers, is formed by two segments of different polymers A and B with a single joint per chain. Most diblock copolymers exhibit a disorder to order transition upon cooling, and this is normally referred to as Upper Disorder-Order (UDOT). The reason for this phase separation upon cooling results from an increase of the interaction energy between the two chemically distinct segments, quantified by χ, the Flory-Huggins segment-segment interaction parameter. However, there are diblock copolymers that undergo microphase separation upon heating.2 The transition from disorder to order observed upon heating is normally named as Lower Disorder-Order Transition (LDOT). It has been shown that, whereas the UDOT is enthalpically driven, the LDOT may arise from differences in volume expansion between the constituent species at a)Author to whom correspondence should be addressed. Electronic mail:
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elevated temperatures, and therefore, it is entropic in its nature. This entropically driven ordering transition is accompanied by a positive change in volume upon demixing of the copolymer blocks.3 Due to this, phase separation can be avoided by pressurization,4 and these polymers are sometimes termed baroplastics. The balance between the entropic and enthalpic behavior has been postulated to be responsible for the peculiar phase separation of these systems5 that in some cases gives rise to a loop behavior where one can find a temperature window of miscibility, and above and below this window, the system is in two phases. Similar behavior has been found in particular blends, where the difference in compressibilities produces a phase separation upon heating.6 Extensive theoretical work has incorporated finite compressibility into the Random Phase Approximation (RPA) through effective interactions.7,8 In that way, χ was reinterpreted as χ = χapp + χcomp, where χapp is the conventional exchange energy with proper density dependence and χcomp represents compressibility difference between constituent blocks. Block copolymers with large compressibility difference and, thus, large χcomp exhibit a gain of entropy due to a volume increase upon phase separation. In any two polymer components A and B, either blends A/B or A-B block copolymers, composition fluctuations are behind the transition from the homogeneous state to the phase separated one, and this is evidenced by a maximum in the static structure factor S(q) at a finite value of the wave vector modulus q∗.9
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From the dynamics point of view, mean field theory for incompressible monodisperse systems10,11 predicts that these composition fluctuations relax via an internal chain mode or breathing mode with a relaxation rate that is independent of q. This mode arises from the relative translational motion of the center of mass of the blocks. For systems with heterogeneity in the composition along the block copolymer chains, Semenov et al. extended the formalism12 and predicted additional composition fluctuations in the form of another mode with a diffusive character, i.e., having a q2 dependence of its relaxation rate. This mode has been referred as the heterogeneity mode, and it is related to the self-diffusion coefficient of the block copolymer. Experimentally, techniques such as light photon correlation spectroscopy (LPCS) allow measuring directly the dynamic structure factor, by using time correlation functions. The scattered intensity correlation function, g2(q, t), can be obtained as ⟨I(q, 0)I(q,t)⟩ ⟨I(q,t)⟩2
.
(1)
In the decade of the 1990s, intensive work proved the existence of the internal mode13,14 and the heterogeneity mode12,15 in UDOT block copolymer melts and solutions16,17 mainly by LPCS. This technique, however, addresses the dynamic structure factor at length scales of the order of 500 nm.18 The use of highly brilliant and partly coherent X-rays, made possible in third generation synchrotrons, allows to use X-rays as a probe to observe the microscopic dynamics at the nanoscale by X ray photon correlation spectroscopy (XPCS). Because the X-ray wavelengths are small compared to those of visible light, XPCS is sensitive to the dynamics at much smaller scales. Recently, XPCS has allowed direct measurements of antiferromagnetic domain fluctuations in ferromagnets,19 slow dynamics in aerogel due to pore breathing,20 or the fluctuations originated in an crystallizing system.21 Related to concentration fluctuation in block copolymers, a few articles have been published in the last years where the dynamics of UDOT diblock copolymers in the disorder state has been studied,22–24 as well as quenched into the ordered state. The direct relation between S(q,t) and the composition fluctuations derived from the mean field theory12 is not valid for systems with finite compressibility, and therefore, the predictions for the existence of the internal mode and the heterogeneity mode prior and during phase transition do not necessarily apply. In this work, we present the first experimental evidence of the existence of two similar modes to those predicted by the mean field theory.9,11–13 By applying XPCS on a LDOT block copolymer, we observe two modes, a q independent internal mode and another with a q2 of the relaxation rate, similar to the ones predicted and observed for incompressible UDOT systems, but presenting some peculiarities by the fact that the system is a LDOT.
Inc.). Films of the sample were prepared by solving in toluene the as received powder and casted into glass substrates. In order to remove the solvent, samples were evacuated at 320 K for two days. Small Angle X Ray Scattering (SAXS) measurements were performed either by a Bruker Nanostar SAXS instrument or at the beamline Spanish CRG BM16 at the European Synchrotron Radiation Facility (Grenoble, France). The lab based Nanostar uses Cu Kα radiation (1.54 Å) produced in a sealed tube. The scattered intensity was corrected with the transmission of the samples calculated considering the absorption of the sample and that of the aluminum encapsulation. A glassy carbon standard was used for this purpose. In the case of measurements at the BM16, a complete description of the beamline setup can be found elsewhere.25 SAXS measurements were made placing the samples either between aluminum or Kapton® foils. The wavelength used in this case was 1 Å. All the SAXS patterns were normalized by the beam intensity, and an empty cell measurement was subtracted. The 2D isotropic SAXS patterns were azimuthally integrated to derive the SAXS curves as a function of the scattering vector q, q = 4π/λ sin(θ),
(2)
where λ is the wavelength and 2θ is the scattering angle. XPCS experiments were performed isothermally at different temperatures. At each temperature, series of above 400 SAXS patterns were collected, maintaining the total exposition time below 25 s to ensure that the sample was not damaged by radiation. Different spots on the sample were illuminated at every temperature. The experiments were carried out at P10 coherence beamline, PETRA III (DESY, Hamburg, Germany). The sample was placed under vacuum. The scattered intensity is recorded on a 2D Maxipix detector (developed at the European Synchrotron Radiation Facility, ESRF, Grenoble, France) with a pixel size of 55 × 55 µm2, 516 × 516 pixel active area, 13 bit/pixel dynamic range, and readout time of less than 0.5 ms. The experiments were performed at 7 keV and the sample to detector distance was 5 m. For each image sequence, the normalized intensity time autocorrelation function g2(t) was calculated pixel by pixel and finally averaged over pixels within a given range of q vector values using the XPCSGUI software.26 Oscillatory shear measurements were performed in a TA Instruments AR1000 rheometer, using samples obtained from toluene casted films. A 25 mm parallel-plate geometry was employed and the gap size was maintained at 0.3 mm. Isothermal frequency sweeps over four decades of frequency (0.01 Hz-100 Hz) were carried out at increasing temperature starting from 373 K to 463 K. A torque value within the linear viscoelastic region (torque = 100 µN m) was applied for all the measurements. III. RESULTS AND DISCUSSION
II. EXPERIMENTAL
The LDOT system studied here is a diblock copolymer of Polystyrene (PS) and Poly(Ethyl Methacrylate), PS-b-PEMA (Mn = 119k, PDI = 1.14, φPEMA = 0.55, Polymer Source,
The evolution with temperature of the SAXS intensity from initially disordered PS-b-PEMA is shown in Figure 1. At low temperatures, the system exhibits a broad peak in SAXS centered around values q∗ = 2π/d, where d = 1.86Rg, being
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FIG. 1. SAXS intensity during heating the initially disordered (not phase separated sample) from T = 390 K to T = 460 K. Measurements taken at BM16, ESRF.25 The inset shows the inverse of the intensity at the maximum (open circles) and full width at half maximum (filled squares) as a function of the inverse temperature during heating. Dotted lines indicate the TODTTDOT.
Rg the radius of gyration of the polymer chain. As temperature increases above 430 K, this peak becomes narrower and more intense, and its maximum shifts towards lower q values. The evolution of the peak characteristics, reciprocal intensity at the position of the maximum, I(q∗), and full width at half maximum (FWHM) as a function of the inverse temperature is presented in the inset of Figure 1. These results suggest that microphase separation is taking place, and from Figure 1, one can estimate the temperature (TDOT) at which the disorder-order transition occurs. For this particular system, we have previously reported a TODT TDOT = 430 K.27 The phase transition can be also corroborated by the development of higher order peaks in the static structure factor, particularly, a second and third order of the main peak, that indicates the formation of a lamellar structure can be observed, Figure 2. The temperature dependence of the interaction parameter for this system with slightly lower molecular weight was reported in the past by Ryu et al.5 The authors showed that χ for this system increases with temperature. In order to understand the dynamical origin of the phase separation observed by SAXS upon heating, XPCS experiments were performed isothermally at different temperatures, from 410 K to 450 K. The obtained autocorrelation function presents, at each temperature, a decay shape that can be modelled with Eq. (3), g2 (q, ∆t) = A + β · exp (−2[t/τ]α ) ,
FIG. 2. SAXS pattern from the “as cast” sample at room temperature (black) and at T = 450 K.
In the studied temperature range, upon heating from the homogeneous state, g2(q, ∆t) reveals a relaxation, with characteristic times in the range from 0.1 to 1 s (open symbols in Figure 3, top). In the bottom of Figure 3, the relaxation time τ, obtained from fitting of Eq. (3) to the relaxation curves, is presented as a function of temperature. Upon heating up to temperatures close to the TDOT, the observed relaxation becomes slower as temperature rises, as revealed by the observed increase in the relaxation time τ. However, as the TDOT is surpassed, the relaxation tends to speed up as temperature increases. These
(3)
where 1/τ and α are the relaxation rate and an empirical exponent that accounts for deviations from a simple exponential. When the value of α is between 1 and 2, the decay function takes the form of a compressed exponential, whereas for values between 0 and 1, the decay function is a stretched exponential. The factor β is a contrast factor that depends upon the beamline conditions and the detector used, and A is a fitting factor very close to unity. The value of A represents the contrast of the speckles, which is mainly determined by the coherent length of the X-ray and the pixel size of the detector.28,29
FIG. 3. Intensity-intensity autocorrelation function g 2, as a function of the elapsed time, obtained at q ∗ value (position of the maximum in SAXS) at different temperatures, upon heating from the homogeneous state (open symbols), and during subsequent cooling (filled symbols) ( ⃝) 410 K, (△) 420 K, () 430 K, (♦) 440 K, (✩) 450 K. In the bottom panel, the characteristic relaxation time is presented as a function of temperature.
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FIG. 4. Intensity-intensity autocorrelation function g2, as a function of the elapsed time, obtained at different q values upon heating from the homogeneous state, at temperatures below DOT (a, T = 410 K) and above DOT (b, T = 430 K). ( ⃝) q = 0.014 Å−1, (•) q = 0.015 Å−1, (△) q = 0.016 Å−1, and (N) q = 0.018 Å−1. Continuous lines in the both panels correspond to best fit to Eq. (3).
results indicate that there might be a change in the nature of the dominant relaxation in the proximity of the disorderorder transition. In fact, the relaxation exhibits a different q behavior in the two different temperature ranges. Figure 4 shows the correlation function for different q values at two selected temperatures, below and above TDOT . Whereas for temperatures T < TDOT, Figure 4(a), g2(t) is independent of the scattering vector and all the curves collapse into a single one, at temperatures T > TDOT, Figure 4(b), the characteristic relaxation time is smaller for higher q values. This can be easily observed by plotting the values of the relaxation rate 1/τ as a function of q2 for different temperatures, Figure 5(a). Below TDOT , in the disordered state, the relaxation rate does not depend on q2, indicating a nondiffusive character of the observed relaxation. This feature is what is predicted for the internal mode of the mean field theory for incompressible diblock copolymers.12 The peculiarity observed in the present case, is that, contrary to what is observed in block copolymers with UDOT,22,23 as temperature increases the relaxation rate decreases. This slowing down is similar to the one reported for UDOT systems when quenched below the TDOT and allows to transit from disorder to order.22 It is worth mentioning that the exponent parameter α in Eq. (3) takes values above 1 in this temperature range. This compressed exponential behavior has been observed for a variety of soft matter systems undergoing jamming transitions, which results in arrested solid like collective dynamics.30–32 As the temperature surpasses the estimated temperature of the transition, the relaxation becomes diffusive with a clear q2 dependence of 1/τ. The predominant mode in this regime, therefore, has the features of that predicted for incompressible
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FIG. 5. Relaxation rate 1/τ as a function of q 2. Continuous lines are linear fits. (a) Heating scan. (b) Cooling scan.
block copolymers with some polydispersity, named in the literature as polydispersity or heterogeneity mode12 and might be related to interdiffusion processes at the interphases.33 The fact that the diffusive motion does not appear in the disordered region, below the TDOT , it might be explained by considering that, in this region, the dynamics is governed by concentration fluctuations that become more intense as the TDOT is approached. The compositional profile may have a weakly sinusoidal shape that becomes more intense as the TDOT is approached. As TDOT is crossed and the system enters a stronger segregation region, these fluctuations disappear and the composition profile evolves from a weakly sinusoidal profile, to a step function, with decreasing width of the interphase. In this situation, the observed relaxation corresponds to the diffusion of chains across the interphase leading to order improvement. LDOT block copolymers allow checking about the persistence of either of the modes in the ordered phase in a wide range of temperatures. This is so because, upon cooling from the ordered state to T < TDOT , the system thermodynamically should be disordered or mixed. However, the ordering prevails as has been observed recently by SAXS27 due to the dynamical arresting produced by the slowing down of the dynamics as the Tg is approached. In our case, the highest Tg block is the polystyrene component, with a Tg ∼ 363 K. The relaxation revealed by the autocorrelation XPCS signal during cooling is presented as filled symbols in Figure 3. The results show that, upon cooling, there is a relaxation process, even at temperatures below the TDOT , with no significant singularity when crossing the transition temperature. Analysis of the q dependence of this process on cooling indicates that it is diffusive along the whole range of temperature studied, exhibiting a q2 dependence of 1/τ (see Figure 5(b)). From the slope of 1/τ vs q2, the chain self-diffusion coefficient (Ds , 1/τ = D s q2) can be obtained. Figure 6
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J. Chem. Phys. 142, 064904 (2015) TABLE I. Relaxation time obtained from XPCS at q = qmax and terminal relaxation time estimation from rheology. T (K) 430 440 450
FIG. 6. Chain self-diffusion coefficient as a function of the inverse temperature of the heterogeneity mode during heating ( ⃝) and cooling (N) scans. The red dotted line indicates the temperature of the disorder-order transition, and the label 600 s indicates the annealing time at T = 450 K.
presents the dependence of D s as a function of the inverse temperature for the diffusive process in both the heating and the cooling scans. During heating, where, according to previous temperature resolved SAXS experiments27 at temperatures above TDOT , the system undergoes progressively the ordering or phase separation process, Ds is higher than the value obtained for the same temperatures during the cooling scan. These results indicate that the better is the order of the structure the slower the diffusion becomes. During the cooling scan, the sample has a quenched lamellar structure and, as revealed by the SAXS patterns, is isotropic. Because of that the obtained diffusion coefficient shown in Figure 6 is isotropically averaged. D s in the lamellar morphology exhibits a curved dependence that resembles a Vogel Fulcher Tamann dependency. When the lamellar structure is formed, the joints between the two different species are likely to be located at the interphase between lamella.
τXPCS (s)*
τtr (s)
0.84 0.67 0.60
18.1 13.2 13.1
Diffusion of a chain implies dragging one block into a region rich on the other component, and therefore, it will experience a thermodynamical penalty. This might be the reason for the retardation compared to the disordered or mixed state. These results are in accordance with recent Monte Carlo simulations based on both the slithering-snake dynamics and the slip-link dynamics,34 where it has been shown that self-diffusion is coupled in the parallel and perpendicular directions, and they predicted a significant slowing down of the dynamics in the lamellar phase compare to the homogeneous melt. Rheology experiments presented in Figure 7 of the sample at selected temperatures after time-temperature-superposition (tT) of the high-ω data show that there is a clear change in the behavior below and above the TDOT . At low temperatures, the experimental conditions do not allow to reach the terminal regime due to the proximity of the glass transition of the blocks. However, as temperature is increased, the rheology of the system start changing and time-temperature superposition does not apply. In the temperature range 430 < T(K) < 450, the system evolves, which can be attributed to the occurrence of the disorder to order transition. At high temperatures, a low-frequency power-law behaviour is found for both G′ and G′′ so that G′ ∼ G′′∼ ω0.5 which is characteristic of ordered lamellar block copolymers.4,35 Thus, confirming the ordered state found by SAXS experiments. At T > TDOT, an estimation for the relaxation time can be obtained from rheology experiments by τtr = G′/(ωG′′)]|ω→ 0.35 The values of the relaxation time, at T > TDOT during the heating scan, obtained from rheology and from XPCS are presented in Table I. Although following the same trend, the values for the terminal relaxation are 1.5 order of magnitude higher than the obtained from XPCS.
IV. CONCLUSIONS
FIG. 7. (a) Storage and (b) loss moduli as a function of the reduced frequency, aTω, for the initially disorder diblock copolymer. ( ⃝) 410 K, (△) 420 K, () 430 K, (♦) 440 K, (✩) 450 K.
We have studied the development of concentration fluctuations in a disordered LDOT diblock copolymer and the diffusive dynamics in the ordered lamellar system by XPCS. Upon heating, before phase separation occurs, a process with the characteristics of the internal mode predicted for incompressible systems is observed, besides the fact that clearly LDOT systems are not incompressible. This mode is peculiar in the sense of exhibiting a slowing down as T increases, which can be understood simply by considering the dependence of the interaction parameter χ with temperature: χ increases with increasing T, due to a competition between the two components contributing to it, an entropic and an enthalpic contribution, and upon heating, the entropic contribution
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becomes more important. Once the TDOT is surpassed, only a diffusion mode is observed. It is worth pointing out again that this diffusive motion becomes slower, but still present, when the lamellar structure is better defined. In this work, we have probed experimentally, for the first time to our knowledge, that the modes predicted for incompressible systems appear also in systems whose phase behavior cannot be explained without assuming compressibility. ACKNOWLEDGMENTS
Financial support from MINECO (Grant No. MAT201123455) is gratefully acknowledged. The experiments performed at P10 in PETRA III (DESY, Germany) were supported by the European Union (Nos. I-20110191 EC and I-20120060 EC). 1M.
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