Strong Memory Effect of Crystallization above the Equilibrium Melting

We attribute the increase in Tc to a strong crystallization memory in the melt above ... The Role of Melt Memory and Template Effect in Complete Stere...
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Strong Memory Effect of Crystallization above the Equilibrium Melting Point of Random Copolymers Benjamin O. Reid,1 Madhavi Vadlamudi,1,‡ Al Mamun,1 Hamed Janani,1 Huanhuan Gao,2 Wenbing Hu,2 and Rufina G. Alamo1,* 1

Department of Chemical and Biomedical Engineering, FAMU-FSU College of Engineering, 2525 Pottsdamer St, Tallahassee, Florida 32310-6046, United States 2 Department of Polymer Science and Engineering, State Key Laboratory of Coordination Chemistry, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing, China, 210093 S Supporting Information *

ABSTRACT: We report the effect of molecular weight and comonomer content on melt crystallization of model random ethylene 1-butene copolymers. A large set of narrowly distributed linear polyethylenes (PE) was used as reference of unbranched molecules. The samples were crystallized from a melt state above the equilibrium melting temperature and cooled at a constant rate. The exothermic peaks of the meltsolid transition are reported as the crystallization temperatures (Tc). Following expectations, the Tc of unbranched PE samples was constant and independent of the initial melt temperature. The same independence was observed for copolymers (2.2 mol % ethyl branches) with molar mass below 4500 g/mol. Moreover, the Tc of copolymers with higher molar mass depends on the temperature of the initial melt, Tc increases as the temperature of the melt decreases. We attribute the increase in Tc to a strong crystallization memory in the melt above the equilibrium melting, and correlate this phenomenon with remains in the melt of the copolymer’s crystallizable sequence partitioning. Albeit molten, long crystallizable sequences remain in the copolymer’s melt at a close proximity, lowering the change in free energy barrier for nucleation. The residual sequence segregation in the melt is attributed to restrictions of the copolymer crystalline sequences to diffuse upon melting and to reach the initial random topology of the copolymer melt. Erasing memory of the prior sequence selection in copolymer melts requires much higher temperatures than the theoretical equilibrium value. The critical melt temperature to reach homogeneous copolymer melts (Tonset), and the comonomer content at which melt memory above the equilibrium melting vanishes are established. The observed correlation between melt memory, copolymer crystallinity and melt topology offers strategies to control the state of copolymer melts in ways of technological relevance for melt processing of LLDPE and other random olefin copolymers.

1. INTRODUCTION The major features of polymer crystallization were uncovered many years ago in the pioneer works of Storks1 and Keller.2 Linear polymers usually develop lamellar crystallites with the chain axis preferentially oriented normal to the basal plane and with thicknesses much lower than the contour length of the molecule. Hence, the polymer molecules must fold back and forth and, except for the very low molecular weight polymers, traverse a crystallite many times connecting two or more lamellae. This molecular connectivity is a unique feature of polymer crystallization. It defines the amorphous or liquid-like region where loops, knots and segments of the chain topologically unable to reach or diffuse to the growing crystal remain trapped; as well as the interfacial region in which the order of the chains emanating from the crystalline phase is dissipated.3,4 Compared to the linear chain, the crystallization of random copolymers with the counit excluded from the crystalline regions is hindered by kinetic constraints associated with © XXXX American Chemical Society

selection and transport through the entangled melt of suitable crystalline sequences.5−8 The long sequences of the distribution are most likely to form the initial nuclei and to crystallize in the early stages of the selection process; other sequences of sufficient length are further pulled to the crystal site where the longest sequence was pinned to, while the shortest sequences remain uncrystallized. This picture of stepwise crystallization due to the sequence selection process generates an intercrystalline region topologically frustrated from the impossibility of dragging to the crystal front all possible sequences, even if of a kinetically suited length. A complex topology builds up in the interlamellar region of random copolymers, especially at the higher levels of transformation, which is associated with their unique crystallization pathway. Sequence selection during crystallization is unique to the random copolymer chain, and Received: April 24, 2013 Revised: July 29, 2013

A

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melt randomization from the fact that melt memory disappeared after increasing holding time in the melt.20,23,30,33 Above a critical holding time, the crystallization kinetics become reproducible and basically independent of melting time.17,23,24,33 All these experimental observations are understood from the basis of thermodynamic phase behavior. At any temperature below equilibrium, a polymer melt is undercooled, and the possibility of finding some crystallites at any temperature between Tmo and the observed melting cannot be ruled out. Furthermore, even at temperatures above equilibrium, segmental dynamics associated with randomizing the melt topology may also be of concern, raising the need for increasing melting time to ensure melt homogeneity. Experiments associated with profound effects of temperature of the melt, and rate of cooling on subsequent crystallization have been carried out mainly with homopolymers.20,23−27,31−33 We are aware of only three studies that analyzed the effect of melt memory in commercial random ethylene copolymers. 29,30,34 In two of these works, the increase in crystallization rate with decreasing melt temperature of ethylene 1-octene copolymers vanished above a critical melt temperature below equilibrium.29,30 Faster kinetics with decreasing melt temperature were also observed by Cheng et al. in two ethylene copolymers, and were attributed to molecular segregation.34 Our interest in the present work, is to further study the effect of the topological structure of the copolymer melt on crystallization rate. We are especially interested in probing if melt-memory in random ethylene copolymers follows the dictates observed in the homopolymer, or if the need for selecting long crystallizable sequences within the entangled random coil adds additional segmental constraints in the phase structure that prevent fast homogenization upon melting. The result of the latter will be a melt structure with memory of the initial sequence segregation needed to build the copolymer crystallites. We demonstrate in the present work that a strong melt memory in random copolymer melts is observed at temperatures even above equilibrium, we discuss the origin of the memory effect, and establish the critical value for temperature of the melt above which crystallization kinetics are reproducible.

is absent in the crystallization of homopolymers. On cooling from the melt, the final copolymer crystal structure is a metastable one with a distribution of crystallites thicknesses and a crystallinity level under evolution toward an equilibrium state.9 While the evolution of the crystalline state is understood on general grounds, the role of the topologically frustrated intercrystalline region on achieving the equilibrium state upon melting has been more elusive. Of more recent concern has been controlling the entanglement density in melts of ultrahigh molecular weight polyethylenes.10−12 The work of Rastogi et al.13 has demonstrated that randomization upon melting of the crystalline structure of high molar mass chains is strongly influenced by the initial semicrystalline morphology.10,11 The variation of melting with annealing time10 and heating rate11 displays different slopes, associated with different activation energies. Crystals formed from nascent disentangled chains have more adjacent folds and less intercrystalline topological ties, hence, melting occurs by chain detachment that further needs to diffuse through the viscous melt. The authors view this melt-state as of a heterogeneous nature, given the likelihood that the detached segments from adjacent chain folds may remain in a more disentangled state than the corresponding to the equilibrated entangled melt. Conversely, in crystals formed from entangled melts, the molecules participate in many crystallites requiring larger co-operative motion to reach the random coil state. Melting of this type of crystallites is likely to involve clusters comprising both, crystals and the entangled intercrystalline regions, rather than via individual chain detachment. Upon melting, the melt-state is homogeneous in reference to having a uniform distribution of entanglements.14 Rastogi et al. have concentrated their efforts in understanding melting and the dynamics of the first order transition toward specific distribution of entanglements;10−15 however, our interest here is in the effect of the topology of the melt on crystallization. This is especially important for random ethylene-1-alkene copolymers with comonomer units excluded from the crystallites. It is well-known that any homopolymer or copolymer melt with clusters of molecular segments with a more ordered conformation than the corresponding to the equilibrated random coil, will crystallize at faster rates due to a depression in free energy change for nucleation from entropic considerations. The source of melt domains with residual order may be from traces of previous crystallites, from oriented molecular segments of sheared melts that did not relax toward the fully equilibrated random state, or regions of less entangled polymer chains that remain at temperatures above melting.16−24 In all cases, melts of this nature display “memory” of a prior ordered state that will affect in some way a subsequent crystallization. Most frequently, melt-memory is observed at temperatures just above the observed melting, or well-below the equilibrium value (Tmo), resulting in higher crystallization temperatures or higher rates of crystallization,18−20,25−30 smaller spherulites and even modifications of the crystal lattice in systems that display polymorphism.26,27,31,32 In most literature works, reproducible and equivalent crystallization kinetics were found from melts above the equilibrium melting temperature, following expectations.20,23−25,29,30,33 When the effect of melt memory on crystallization rate was found upon cooling from temperatures above equilibrium, it was most often associated with incomplete

2. EXPERIMENTAL SECTION 2.1. Materials. Three sets of samples were used in this study. The first set, used as a reference for the unbranched chain, are narrowly distributed linear polyethylene fractions covering a molar mass range from 4 × 103 to 8 × 105 g/mol and Mw/Mn < 1.5. These fractions were obtained prior to 1995 from the Société Nationale des Pétroles d’ Aquitaine (SNPA) in France and have been properly stored. The molar mass characteristics obtained from GPC for the linear polyethylene fractions are listed in Table 1. Also listed are peak crystallization and melting temperatures obtained from differential scanning calorimetry after cooling and heating at 10 °C/min. The second set of samples consists of hydrogenated polybutadienes (HPBD), which are analogous to random ethylene 1-butene copolymers, and serve as models for crystallization studies. The molar mass characteristics of the HPBD samples are listed in Table 2. These samples have ∼2.2 mol % ethyl branching and cover a molar mass range from 800 to 420 000 g/mol. The HPBD samples were synthesized by ionic polymerization and have a narrow molar mass distribution (Mw/Mn ∼ 1.1). P120* is a 3-arms star HPBD that is equally weighted (40 000 g/mol) in each arm and contains the same ∼2 mol % ethyl branching randomly distributed on each arm. The samples of the third set are HPBD with molar mass between 50 and 100 kg/mol, and ethyl branching content varying from 2 to 5.68 mol %. The molar mass characteristics and branching content are B

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Table 1. Characterization of Linear Polyethylene Fractions (SNPA Samples) sample ID

Mw (g/mol)

SNPA4K SNPA10K SNPA16.5K SNPA69K SNPA102K SNPA189K SNPA316K SNPA800 K

4 10 16 68 102 189 316 911

000 000 500 700 000 000 000 000

PDla

Tcb (°C)

Tmb (°C)

1.06 ∼1.1 1.26 1.32 1.43 1.11 ∼1.1 1.3

112.0 116.0 115.7 115.3 114.4 113.9 114.1 110.6

126.5 128.7 132.2 131.0 131.9 130.7 130.2 124.4

a

Polydispersity index (Mw/Mn). bPeak temperatures from DSC cooling and heating at 10 °C/min.

Table 2. Characterization of Hydrogenated Polybutadiene Samples (HPBD) with ∼2.2 mol % Ethyl Branching sample ID

Mw (g/mol)

P0.8 P2.2 P4.53 P6.95 P16 P24 P49 P79 P108 P120*c P145 P420

2 4 6 16 24 49 79 108 120 145 420

800 200 530 950 000 000 000 000 000 000 000 000

PDla

ethyl branching (mol %)

Tom copo b (°C)

∼ 1.1 1.1 1.1 1.1 1.4 1.1 1.1 1.1 1.1 ∼1.1 1.1 1.84

∼2.2 2.16 2.27 2.36 2.10 2.30 2.22 2.44 2.20 2.2 2.7 2.22

137.8 137.8 137.4 137.1 138.2 137.5 137.7 137.0 137.7 137.8 135.9 137.7

Figure 1. DSC thermal protocol for melting and cooling runs at 10 °C/min. The horizontal line corresponds to the equilibrium melting temperature of a random copolymer with 2.2 mol % branches. It is added to indicate that the temperatures of the melt, prior to cooling, fluctuate above and below this value in a nonsystematic manner. cooling exotherms. The melting endotherms were also analyzed to verify that the melting temperature and latent heat of fusion were independent of the initial melt temperature, thus ensuring the same type of crystal formed for every trial. As shown in the thermal protocol of Figure 1, the temperatures of the initial melts for each cooling trial were varied in a random pattern, above and below the equilibrium value, to avoid any possible systematic errors in the measured crystallization temperatures. For each sample, a second trial was performed where the temperature of the melt was reached from above, i.e. by cooling from an elevated temperature to Tmelt, holding for 5 min at Tmelt and then cooling to 40 °C at 10 °C/min. Furthermore, trials were also carried out to probe time requirements for chain diffusion in the melt. These trials involved additional time in the melt, and were performed by holding the samples for 10, 30, or 60 min at Tmelt before cooling. The morphology was recorded using an Olympus BX51 optical microscope with polarized light, and equipped with an Olympus DP72 digital CCD camera. The temperature was controlled using a Linkam hot stage with a temperature programmer (Type TMS94), also commercialized by the Linkam Scientific Instruments Ltd., U.K. Simultaneous wide and small-angle X-ray diffraction patterns at elevated temperatures were obtained using a Bruker Nanostar diffractometer with Iμs microfocus X-ray source (λ = 1.5412 Å), and equipped with a HiStar 2D Multiwire SAXS detector and a Fuji Photo Film image plate for WAXD detection. The plates were read with a Fuji FLA-7000 scanner. The diffractometer was equipped with a Nanostar type H hot stage and the temperature was controlled by Nanostar TCPU H controller. Prior collection of WAXD or SAXS patterns, all thermal history was erased by bringing the sample to 200 °C, cooling to 40 °C and then heating to the desired temperature. The heating and cooling rate was 10 °C/min.

a

Polydispersity index (Mw/Mn). bCalculated according to eq 1. c3arms star HPBD. listed in Table 3. The HPBD samples, all used in prior works,5,6,35−37 were supplied by Dr. William W. Graessley38,39 and by the Phillips

Table 3. Characterization of Hydrogenated polybutadiene samples (HPBD) with different content of ethyl branching sample ID P108 ArgP3.6 P98 P97 a

Mw (g/mol) 108 108 50 50

000 000 000 000

PDla

Ethyl Branching (mol %)

T0m copob (°C)

1.1 1.1 1.1 1.1

2.20 3.60 4.14 5.68

137.7 132.9 131.0 125.7

Polydispersity index (Mw/Mn). bCalculated according to eq 1.

Chemical Co. Sample ArgP3.6 was kindly donated by Prof. Carella from National University of Mar del Plata, Argentina (INTEMA). The unbranched linear polyethylene fractions and the HPBDs were studied using the same thermal history protocols. Films, about 150 μm thick, were prepared from the original powders by melting between Teflon foil in a Carver press at 150 °C under low pressure and quenching at room temperature. To ensure good contact, a single flat piece, ∼4 mg, was cut from each sample and placed in DSC aluminum pans. The crystallization and melting behaviors of the polymers were carried out in a Perkin-Elmer differential scanning calorimeter (DSC-7), using Pyris software. Some experiments were carried out in a TA Q2000 DSC. The instruments were calibrated with Indium, and the calibration checked every day. The main thermal protocol used is schematically shown in Figure 1. The samples were heated from 40 °C up to a temperature well above the observed melting (Tmelt) at a rate of 10 °C/min. After 5 min had elapsed, the samples were cooled to 40 °C at a rate of 10 °C/min. The crystallization temperatures were recorded from the peak of the

3. RESULTS AND DISCUSSION 3.1. Crystallization of Unbranched Linear Polyethylene Fractions. The effect of melt temperature on the crystallization kinetics of linear polyethylene fractions was analyzed by Ergoz et al.33 It was found in this work that fractions with molar mass below 1 × 106 g/mol have reproducible isothermal kinetics when crystallized from melts that were held for 10 min at Tmelt ≥ 140 °C. Higher molar mass fractions required longer holding times at 140 °C (≥50 min) or higher melt temperatures to reach similarly reproducible kinetics. These early data made relevant the need to increase holding time of highly viscous melts to allow for chain diffusion and to reach homogeneity of the random molten system. C

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Figure 2. Plot of heat flow against temperature for cooling from melt temperatures of 200, 180, and 150 °C and subsequent heating at 10 °C/min for SNPA16.5K (left) and SNPA316 K (right). A dashed line is drawn at the equilibrium melting, Tmo = 145.5 °C. Exothermic peaks point down, endothermic up.

Furthermore, the critical temperature of 140 °C to reach a homogeneous random melt is below the equilibrium melting temperature for polyethylene, taken as 145.5 °C,40 indicating that linear polyethylene even of a very high molar mass is free of melt memory at temperatures above equilibrium, and behaves similarly to other homopolymers.20,23−27 Melt memory in linear polyethylene is found at temperatures below Tmo and is particularly prominent for high molar mass fractions. Values of Tmo for linear polyethylenes lower than 145.5 °C have been proposed,41,42 but the majority are above 140 °C and hence do not change the fact that memory, when present in homopolymers, is restricted to undercooled melts. The early data by Ergoz et al. obtained by dilatometry are contrasted with DSC data of crystallizations carried out following the thermal protocol of Figure 1 on the large number of narrow fractions listed in Table 1. These fractions cover a range in Mw from 4000 to 910 000 g/mol. Representative examples for SNPA16.5K and SNPA316K are shown in Figure 2. The figures display identical cooling and heating cycles from three different melt temperatures, all above Tmo (shown as a vertical dashed line in the figures). All other cycles starting at Tmelt > 145.5 °C led to identical superposed thermograms. The crystallization and melting thermograms lie on top of each other signifying crystallization kinetics and melting processes independent of the initial Tmelt ≥ 145.5 °C. Only when Tmelt falls below Tmo and approaches the range of isothermal crystallization temperatures experimentally accessible, the cooling exotherm is displaced to a higher temperature range characteristic of a faster crystallization. The peak crystallization temperatures (Tc) are plotted as a function of the temperature of the melt (Tmelt) in Figure 3. The dotted horizontal line in this figure represents the equilibrium melting temperature, drawn as a reference to the thermodynamic melting value. The data of Figure 3 first confirm the molar mass dependence of the crystallization rate, associated with the value of Tc, as amply demonstrated in prior works.33,43−47 Except for the influence of chain ends that decrease the crystallization rate in the lowest molar mass fraction (4K), the rate decreases with increasing chain length independently of Tmelt, as found by the progressively lower values of Tc with increasing molar mass. A second point of interest in Figure 3 is that for Tmelt > Tmo, the Tc of any given fraction falls on the same value, represented by the continuous straight lines, indicating a constant crystallization rate

Figure 3. Temperature of the initial melt (Tmelt) vs peak crystallization temperature (Tc) for linear polyethylene fractions.

independent of the temperature of the initial melt. Hence, we confirm that Tc is not affected by melt memory when Tmelt > Tmo for the homopolymer as expected, and as it was found in earlier work.33,48 For Tmelt < Tmo, the crystallization temperature deviates in most fractions toward higher values, denoting an enhancement in the rate due to generation of nuclei in the undercooled melt or due to an incomplete melting. The data of Figure 3 for unbranched linear polyethylene fractions serve as a reference to further analyze the crystallization behavior of the random copolymers. The degree of crystallinity obtained from the area of the melting peaks, taking 995 cal/mol as the heat of fusion per mole of crystalline units,35 is plotted as a function of molar mass in Figure 4 (filled circles). In this figure, DSC crystallinity contents are compared with dilatometric data for isothermally crystallized fractions extracted from the work of Ergoz et al.33(open circles). The vertical shift between the two data sets can be attributed to the different methods of crystallization. The crystallinity of isothermally crystallized samples is 10−20% higher than for fractions cooled from the melt at 10 °C/min. However, the same trend is observed in both sets of data, for molar mass between 4K and 20K, the level of crystallinity is the D

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major shortcoming of eq 1 is that the predicted equilibrium values are much higher than any random copolymer experimental melting. This disparity is demonstrated in Figure 5 where the final melting temperatures (Tmf) of hydrogenated

Figure 4. Degree of crystallinity as a function of molecular mass for linear polyethylene fractions: (○) data from Ergoz et al. for isothermally crystallized samples;33 (●) data from this work for samples cooled at 10 °C/min.

highest, ∼80% and decreases to ∼40% with increasing molar mass until about 2000 kg/mol. The level of crystallinity levels off with a further increase of molecular mass. The decrease of level of crystallinity is explained by the increased number of entanglements per chain with increasing molar mass. As molecular weight increases, the knots and entanglements in the melt increase, restricting mass of crystalline material. 3.2. Crystallization of Random Ethylene Copolymers. 3.2.1. Effect of Molar Mass at a Fixed Comonomer Content. The first part of this section examines the effect of molar mass on the melting and cooling characteristics of hydrogenated polybutadienes with a fixed branching content of ∼2.2 mol % following the thermal cycling of Figure 1. Cooling at a constant rate of 10 °C/min is carried out from melt temperatures above and below the equilibrium melting temperature of the copolymers, which is calculated according to Flory’s phase equilibrium theory.9 Flory developed the most rigorous thermodynamic theory for melting of copolymers under the assumption that the minor counit does not enter the crystal lattice.9 When treated as models for ethylene 1-butene copolymers, the minor counit of hydrogenated polybutadienes is associated with 1-butene and the crystallizable unit is ethylene. According to this equilibrium theory, the melting temperature of the copolymer, Tom copo, relative to that of the homopolymer, Tmo, can be expressed as: 1 1 R ln p = o − o Tm Tmcopo ΔHu (1)

Figure 5. Experimental final melting temperatures (Tmf) of hydrogenated polybutadienes plotted as a function of ethyl branching content and thermodynamic equilibrium melting according to eq 1. The line drawn for the experimental points represents the best visual average over these data.

polybutadienes studied by Crist et al.8 and those of this work are contrasted with the values calculated with eq 1. The thermodynamically defined melting temperature of the copolymer, Tom copo, corresponds to equilibrium between the thickest possible crystals, those formed by packing the longest ethylene sequences, and a melt having the global copolymer composition XA. Hence, in Figure 5, we have plotted the final observable melting, rather than the DSC melting peak value, because the final melting is the closest reflection to equilibrium. Only at that point crystallites will coexist with a melt with a composition corresponding to the chain’s comonomer composition. The departure from equilibrium of the observed melting is the inability of the copolymer to achieve the structural conditions stipulated by the equilibrium requirements. The thickest equilibrium crystallites do not form because the number of longest sequences is extraordinarily small, too small to be detected even if they would form; besides, kinetic constraints associated with transporting the required long sequences through the entangled melt make the process unduly long and topologically frustrated. For these reasons, the level of crystallinity that actually develops, and the average thickness of the crystals is less than required by equilibrium, with the concomitant depression in the observed melting, decreasing with counit content, as shown in Figure 5. Apart from experimental difficulties from achieving equilibrium crystallites, the theory establishes that a copolymer melt below Tom copo is undercooled. Crystallites may be formed or may survive at any temperature between the observed Tmf and Tom copo, even if in such small amounts that are not observable. Copolymer crystallization in this temperature range will be difficult and lengthy, more so as the temperature approaches Tom copo, but not impossible as crystallization falls below the equilibrium line. The theory also establishes an equilibrium crystallinity that vanishes sharply at a temperature approaching

In this equation, ΔHu is the enthalpy of fusion per mole of crystalline repeating unit, and p is the crystallizable sequence propagation probability. In statistical copolymers, p < 1 and the equation predicts a large depression of the equilibrium melting of copolymers. For random copolymers, p = XA, the mole fraction of crystallizable units, and Tom copo decreases proportionally to the content of noncrystallizable counits. For an ordered or block copolymer, p approaches 1, thus predicting a negligible melting point depression in the copolymer. The relation between sequence distribution and melting temperature given by eq 1 has been amply verified experimentally,7,35,36,47 and the application of this phase equilibrium concept to the characterization of LLDPEs has been proven to be very useful.49−51 However, for quantitative applications, a E

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Figure 6. Plot of heat flow against temperature for cooling from melt temperatures in a range of 200 to 100 °C and subsequent heating at 10 °C/min for hydrogenated polybutadienes P2.2, P24, P108, and P120*. Differences in crystallization temperatures are magnified in the insets of P24 and P120*. The thermograms of P108 were obtained in a DSC TA2000 with automatic level and zero baseline corrections. All other thermograms were obtained using a Perkin-Elmer DSC7. The dashed line is drawn at the equilibrium melting of the copolymers, Tom copo = 137.8 °C. Exothermic peaks point down, endothermic up.

demonstrate that a 2 −3 °C shift of Tc to higher values occurs with decreasing Tmelt, even at Tmelt > Tom copo. Hence, what we observe is an unexpected crystallization kinetics strongly affected by the state of the initial melt even above Tom copo. This melt-memory feature, absent in homopolymer fractions, is found in all model random copolymers above a threshold molar mass. The range of Tmelt above Tom copo where ethylene copolymers display the effect of melt-memory on crystallization is easily extracted in the plots of peak crystallization temperature vs Tmelt, which for clarity we present in two separate figures. Figure 7(a) displays data for low molar mass copolymers, and Figure 7b data for representative copolymers of higher molar mass. As seen in Figure 7a, for Tmelt ≥ Tom copo, copolymers with molar mass up to ∼5000 g/mol have crystallization rates independent of the initial melt temperature. All Tc fall on the same line in this range, and thus follow the behavior found for the homopolymers. For molar mass >7000 g/mol, the vertical linearity of Tc in Figure 7(b) is maintained for Tmelt > ∼ 160 ± 10 °C, depending on molar mass. For lower Tmelt, Tc shifts consistently to higher values indicative of an enhanced crystallization. Given that 160 °C is above Tom copo and that enhanced crystallization was not found for linear polyethylene fractions above equilibrium, the melt-memory effect on crystallization of these copolymers must be associated with the presence of branches, and therefore, with the redistribution

Tom copo, and the pure single phase melt at temperatures above equilibrium melting. With a clear demarcation for equilibrium melting of random copolymers we can now analyze their crystallization behavior from temperatures above and below equilibrium. Representative cooling and melting thermograms are displayed in Figure 6 for copolymers with the same 2.2 mol % branches and molar mass increasing from 2,200 to 120 000 g/mol. Note that P120* is a symmetric three arms star copolymer. The vertical discontinuous line corresponds to the equilibrium temperature. Differences in the experimental thermograms obtained under multiple cycling are quite striking. The behavior of the copolymer with the lowest molar mass (P2.2) is equivalent to the behavior of the homopolymer fractions. Cooling from melt temperatures above Tmo copo gives for the shortest chains reproducible crystallization kinetics and identical melting endotherms, thus reflecting equilibrated or uniform chain conformations in the initial melt. The situation changes for longer copolymer chains as seen in Figure 6. Dynamic cooling and melting of the other three copolymers shown in this figure display invariant melting temperatures, and crystallization temperatures which increase as the temperature of the initial melt (Tmelt) decreases even for Tmelt above Tom copo. This is the typical behavior of a faster crystallization kinetics associated with melt memory. The insets display exotherms from melt temperatures above Tom copo in a more expanded scale and F

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Figure 8. Polarized optical micrographs of P108: (a) morphology from a homogeneous melt, Tmelt = 200 °C; (b) morphology from a heterogeneous melt, Tmelt = 150 °C. The scale bar represents 20 μm.

previous selection and packing of crystalline sequences. The second issue deals with melt dynamics, or ensuring the copolymer is given enough time at Tmelt to allow any residual order to dissipate back to the homogeneous disordered state. To address the first issue, the samples were heated to a completely disordered state before cooling from a Tmelt that exhibits the increase in Tc when the same Tmelt is approached from below. For example, P16 and P108 were first heated to 200 °C for 5 min (a homogeneous Tmelt free of memory), they were then cooled from 200 to 150 °C, held at 150 °C for 5 min and further cooled to 40 °C to record Tc. Since the thermal history was erased before crystallization, any increase in Tc must be due to preordering in the melt at 150 °C. The values of Tc obtained for P24 and P108 under this thermal protocol are shown as filled squares in Figure 9 together with data from Figure 7. Temperature of the initial melt vs crystallization temperature for the HPBD copolymers indicated. (a) Low molar mass copolymers (800 < Mw < 7000). (b) HPBD copolymers with higher molar mass (16 000 < Mw < 420 000), all with ∼2.2 mol % ethyl branches. The dashed line is drawn at the equilibrium melting of the copolymers, Tom copo = 137.8 °C.

of the branches upon melting. We term the state of these melts heterogeneous, and the state of randomized melts homogeneous. Independently of the initial Tmelt, as the molar mass increases from 16,000 to 420 000 g/mol, there is a clear shift of Tc to lower values, a feature already discussed in prior work5,47 and associated with the effect of increasing melt viscosity in raising the energy barrier for segmental transport, i.e., as the molar mass increases, the polymer molecules become more sluggish. The change in nucleation density that is associated with the increase in Tc in Figure 7b results in a dramatic morphological change as illustrated in the polarized optical micrographs (POM) of Figure 8 for P108. In this figure, the number of nuclei formed on cooling from the homogeneous melt (Tmelt = 200 °C, Figure 8a) is clearly lower than cooling from a heterogeneous melt (Tmelt = 150 °C, Figure 8b), as shown respectively by the coarse vs fine spherulitic morphology of the micrographs. We next address two issues in order to elucidate the origin and nature of self-seeds that survive temperatures above the equilibrium melting point of copolymers and reduce the entropy penalty for nucleation. The first issue is determining if the seeds are left in the melt as a memory from prior crystalline sequences that, although they may have lost crystallographic packing, they remain molten, but in close proximity due to a relatively slow thermal mobility, or if some type of segmental order builds in the melt (above Tom copo) independently of a

Figure 9. Initial melt temperatures vs cyrstallization temperatures for samples P16 (right) and P108 (left). Data approaching Tmelt from above (red solid squares), and data from melts approached from below and held 5 min (solid black circles), 10 min (open triangles) and 30 min (red crosses) are included. The dashed line represents the equilibrium melting temperature of the copolymers.

Figure 7b where Tmelt was approached from below. They fall in the vertical straight line of Tc data for Tmelt > 160 °C, and thus originate from a homogeneous melt. In other words, there is no melt memory at Tmelt ≥ Tom copo when Tmelt is approached from above. This confirms that melt memory originates from the copolymer crystalline structure that evolves during the first cooling to 40 °C. The observed molar mass effect on melt memory is a strong indication that diffusion must be playing a role, hence, to address the issue of time at Tmelt prior to cooling, the copolymers were held for times that ranged between 10 and 30 min at the initial melt temperature. The extra time at Tmelt may G

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Figure 10. (a) Simultaneous wide-angle X-ray diffraction and (b) small-angle X-ray scattering patterns of P108 collected at the temperatures indicated. The inset in part b represents Lorentz-corrected SAXS intensities. Prior collection of WAXS and SAXS patterns, all thermal history was erased by bringing the sample to 200 °C, cooling to 40 °C and then heating to the temperature shown.

between 200 and 110 °C of Figure 11. A change in melt topology with increasing Tmelt leading to two different

allow the copolymer crystalline sequences to diffuse and redistribute back to the fully disordered state.24,52 However, diffusion appears to be a very slow process as the data fall mostly on the 5 min Tmelt line (Figure 9). Even holding for 3 h, the melt of P108 at 150 °C made no significant changes in Tc compared to a 5 min hold of the melt. What is then the exact nature of the self-seeds? Could crystal fragments survive temperatures >30 degrees above the equilibrium melting point? DSC and POM give no indication of any remaining crystallites above Tmo copo. Furthermore, possible remains of crystalline structure were probed by WAXS under the same thermal treatment of Figure 1. As shown in Figure 10a, the diffractograms taken at temperatures ≥110 °C display only the amorphous halo of P108. Crystallographic reflections are absent at Tmelt > Tom copo. Then, crystallites do not survive, as shown, but clusters of molten long ethylene sequences must remain above Tom copo as long-live transients. If any partial order remains within these clusters, it must be very low or the correlation distance between the clusters is beyond the experimental small-angle X-ray (SAXS) collection range, as there are no SAXS peaks and there are no differences in the SAXS patterns at Tmelt above and below Tom copo (Figure 10b). The extensive thermal data, supported by morphological and structural information of Figures 9 and 10, indicate that after ethylene sequences are partitioned to build the crystalline state, prolonged heating to a range of Tmelt above Tom copo destroys the coherence of the crystal lattice but it does not redistribute the sequences and branches at once, as shown schematically in the graphic that accompanies the abstract. Clusters of molten ethylene sequences with little or no branching remain in the melt even above Tom copo as a memory of the crystallizable sequence length segregation in the evolution of the crystallites. The time that may allow diffusion to the homogeneous melt state appears to be unduly long, but not unreasonable. Other works also found that melt temperature has a greater and more dramatic effect on the concentration of active nuclei than time in the melt.24,28 Increasing melt temperature enables sufficient thermal segmental motion to randomize the copolymer melt structure. The demarcation between heterogeneous and homogeneous copolymer melts as Tmelt increases above Tom copo is not sharp, as shown in the expanded cooling exotherms for P108 from Tmelts

Figure 11. Exotherms from cooling P108 at 10 °C/min from melt temperatures (Tmelt) between 110 and 200 °C. Each of the melt temperatures was approached from 40 °C.

populations of nuclei becomes quite apparent in this figure. Nuclei that form from a randomized melt (Tmelt of 200 and 180 °C in Figure 11) develop identical exotherms with a peak Tc = 80.2 °C. In the other extreme, nucleation from a widely heterogeneous melt, such as for Tmelt of 110 and 130 °C where long sequences retain a signature of their partitioning during crystallization, also develops single exotherms but at a significantly faster rate, here the peak Tc is ∼83.2 °C. At intermediate Tmelt, a fraction of the melt remains heterogeneous, while in the remaining melt, sequences possibly from thinner crystallites randomize and can nuclei at a later stage from a homogeneous melt. These two types of nucleation are prevalent in Figure 11, as observed by double peaked exotherms (Tmelt = 150 °C) or broader exotherms covering both nucleation regimes (Tmelt of 140 and 160 °C). In the crystallization exotherm from Tmelt = 150 °C, the peak at low Tc (∼81 °C) is associated with nucleation from homogeneous melt, while the shoulder at higher Tc (∼83 °C) develops from H

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Figure 12. Effect of copolymer crystallinity on melt topology of P108 reflected on last step crystallization exotherms (from 150 to 40 °C). From homogeneous 150 °C melt (runs: ref2. and 200−91(2hr)-150−40) coarse spherulitic morphology develops in polarized optical microscopy, due to low number of nuclei. From heterogeneous 150 °C melt (runs: reference point 1, 200−91(2 h)−40−150−40 and 200-stepwise to 40−150−40) a morphology with fine texture develops due to a high number of nuclei (top image). The scale bar on the lower right corner represents 20 μm.

the sequences that remain clustered in the melt. As Tmelt increases, the fraction of homogeneous melt increases, and consequently, the crystallization shifts toward the uniform nucleation observed at Tmelt above the critical value to reach melt homogeneity. The results on model random ethylene 1-butene copolymers contrast with literature data on copolymers synthesized with a coordination-type catalyst.29,30 A melt memory effect was also observed in commercial ethylene 1-octene copolymers synthesized with a metallocene catalyst but only for Tmelt < Tom copo.29,30 The ethylene 1-octene copolymer with 3.65 mol % hexyl branches studied by Androsch et al.29 displayed enhanced crystallization for Tmelt < 127 °C, while the equilibrium melting temperature for this copolymer is 133 °C. Similarly, the ethylene 1-octene copolymer with 3.9 mol % branches analyzed by Strobl et al.30 showed reproducible crystallization kinetics from Tmelt > 127 °C which is well below the equilibrium value of 132.1 °C for this copolymer. The difference with the present work is attributed to the commercial nature of the copolymers analyzed or to the method of synthesis. In addition to having a broader molar mass distribution, commercial copolymers, or copolymers synthesized with a metallocene catalyst and MAO, contain additives or residues from the cocatalyst that act by themselves as nucleants.53 The nucleating effect from these “foreign” centers may be very high compared to the melt memory effect from crystallizable sequence selection, such that the latter is not resolved experimentally. Indeed, our own data on commercial ethylene copolymers corroborate that additives may be a large contribution to nucleation. For example, a commercial ethylene 1-hexene with an average 1.87 mo% hexyl branches that was subjected to the thermal protocol of Figure 1, did not display the shift toward higher crystallization peaks for Tmelt > Tmo copo. However, after the same copolymer was subjected to a Soxhlet extraction using a 0.8 μm pore size thimble to extract most of the additives, the copolymer did show the typical signature of enhanced crystallization from melts above Tmo copo (Figure S1, Supporting Information). Therefore, we conclude that melt memory of the crystallizable sequence partitioning that evolves during copolymer crystallization is a general feature of fusion of random copolymers. The work of Cheng et al.34 on isothermal crystallization of commercial metallocene ethylene 1-butene (2.1 mol %

branches) and ethylene 1-hexene (0.8 mol % branches) is of interest to the present study because crystallization was carried out from Tmelt = 150 °C, a temperature above the equilibrium value of both copolymers. These authors observed that rapidly crystallized copolymers produced a melt from which a subsequent crystallization is slower than from melts of the stepwise crystallized copolymers. In the stepwise crystallization, the temperature was decreased from 150 °C in 3−5 °C steps and held 12 h for each step to favor sequence segregation. From the analysis of the kinetics it was concluded that the observed faster kinetics was a consequence of molecular rather than sequence segregation. According to these authors, molecular segregation led to phase separation in the melt, which is enhanced by residence time, and driven by intermolecular heterogeneity of the comonomer content.34 Intermolecular composition heterogeneity is inherent to most commercial LLDPE obtained via coordination catalysis,54 and it was proven to lead to some degree of melt phase separation in copolymers synthesized with a Ziegler−Natta catalyst.55 However, comonomer composition heterogeneity is not an issue with the hydrogenated polybutadienes studied in this work, and liquid−liquid phase separation highly unlikely. The anionic polymerization precludes such heterogeneity, and ensures a very narrow molecular weight distribution (Mw/Mn ∼ 1). We tested if the content of long crystallizable sequences that remain clustered in the melt at a fix temperature above equilibrium increases after repeated cooling and melting cycles from the same Tmelt. For this experiment, the memory of P108 was first erased by melting to 200 °C and after cooling to 40 °C, the temperature was eight times raised to 150 °C and cooled to 40 °C following the standard protocol of holding 5 min after each cycle. The crystallization (and melting) peaks were identical for each cycle (Figure S2). There is no segregation ripening and no additive epitaxial effects on the number of nuclei that remain effective in the melt at 150 °C from a subsequent crystallization. 3.2.2. The Role of Copolymer Crystallinity and Melt Topology on Memory above Equilibrium Melting. Enhanced crystallizable sequence partitioning under a more selective crystallization, such as isothermal or stepwise processes, may indeed lead, after fusion, to a more heterogeneous melt, and to I

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constraints in the melt that may slow diffusion of these sequences back to the homogeneous melt. These results clearly provide a close correlation between copolymer crystallinity and melt topology, and offer strategies to control the state of random copolymer melts in processes of technological relevance. In Figure 7, vertical lines are drawn over the Tmelt data that gives a constant Tc, the Tc data associated with a homogeneous randomized melt. With this line as reference, the onset melt temperature for enhanced kinetics (Tonset) is taken as the middle point between the experimental Tmelt that first exhibits an increase in Tc and the previous homogeneous Tmelt (illustrated by points A and B in this figure). Tonset is a critical temperature to reach a homogeneous melt state for random ethylene copolymers with 2.2 mol % branches (ethyl and larger). The change in Tonset with molar mass is significant, as shown in Figure 13, and scales linearly with the logarithm of the molar

an even faster crystallization. This possibility was tested with P108 due to the sensitivity of this copolymer to the two types of nuclei when crystallized from a heterogeneous Tmelt of 150 °C. The copolymer was first brought to 200 °C to erase all memory and further cooled to the isothermal crystallization temperature of 91 °C and held at this temperature for 2 h for complete transformation. At this temperature the t1/2 is ∼20 min, and the crystallinity ∼9%. To test if sequence partitioning after crystallization at 91 °C led to a heterogeneous melt, the temperature was raised from 91 to 150 °C and then cooled to 40 °C to register Tc. There are negligible differences in the last crystallization peak of the sequence 200−91(2 h)−150−40 compared to the reference crystallization from a homogeneous melt, 200−40 (reference point 2 in Figure 12). The spherulitic morphology at the end of both processes is also identical, as shown in this Figure. This result is quite striking, and initially unexpected. It indicates that in spite of the selection of relatively long crystallizable sequences in the isothermal process, the crystallites melt to a homogeneous state when brought to 150 °C. Hence, it appears that reaching a homogeneous melt is associated with the low level of crystallinity that copolymers develop at high crystallization temperatures due to the required crystallizable sequence length.5 This result emphasizes the kinetic, rather than thermodynamic nature of melt memory of random copolymers, which depends more strongly on development of high degree of crystallinity prior to melting than on the formation of the thickest possible crystallites. To further probe the correlation between degree of crystallinity and copolymer chain topology upon melting, P108 was again crystallized isothermally at 91 °C for 2 h and subsequently cooled to 40 °C. In the cooling step the crystallinity content increases from 9 to 30%. Then the temperature was raised to 150 °C and cooled to 40 °C to record the last exotherm. The structure of the melt at 150 °C now has the signature of the heterogeneous melt, as shown by a much faster crystallization (Figure 12). In fact, the crystallization and morphology from the last step of the sequence 200−91(2 h)−40−150−40 is identical to the reference of the heterogeneous melt, 200−40−150−40 (reference point 1 in Figure 12). The observed morphology with a finer texture is also the expected one for crystallization from heterogeneous melts (Figure 12). When the step of isothermal crystallization at 91 °C was replaced by a stepwise crystallization (cooling from 200 °C in 3 deg steps and holding for 30 min in each step), a subsequent heating to 150 °C and cooling to 40 °C led to an identical high Tc exotherm, also shown in Figure 12. We then conclude that the structure of heterogeneous melts is set by the intercrystalline topology acquired in the initial cooling to 40 °C. This conclusion is corroborated by the lack of melt memory when P108 was isothermally crystallized at 91 °C from a homogeneous melt at 200 °C instead of from a heterogeneous melt. The isothermal experiments described above indicate that it is not so much the partitioning of long crystallizable sequences during copolymer crystallization what leads to heterogeneous melts at Tmelt > Tom copo, but the chain topology of loops, links and ties that generate in the intercrystalline regions after a sufficiently large crystallinity content evolves. Even if the long ethylene sequences are partitioned from the homogeneous melt during the slow isothermal crystallization at 91 °C, the ∼9% crystallinity that develops is too low to build sufficient

Figure 13. Critical temperature for melt homogeneity plotted as a function of molecular mass of random ethylene 1-butene copolymers with 2.2 mol % ethyl branches. The discontinuous line represents the equilibrium melting temperature neglecting end effects.

mass for Tmelt > Tom copo. Here, the data for P79 and P145 are not included due to the higher branching. The data of P49 was also excluded from this plot because we suspect this sample contains some impurities that are causing anomalous Tonset. The bars drawn on the data of Figure 13 demarcate the two experimental Tmelt values used to obtain Tonset. They represent a range of uncertainty in Tonset based on the experimental data collected, every 10 degrees for molecular weights ≤16 000 g/ mol and every 5 deg for higher molar masses. The critical copolymer temperature to reach a homogeneous melt state decreases from ∼175 °C to ∼145 °C with decreasing molar mass and intercepts the equilibrium value (Tom copo) at a molar mass of 1300 ± 200 g/mol. Remarkably, this threshold length to prevent melt memory of the crystallizable sequence segregation at Tmelt > Tom copo, is the critical entanglement molecular weight for polyethylenes.56−58 The differences in melt homogeneity at temperatures above equilibrium between homopolymer fractions (Figure 3) and random copolymers with fixed comonomer content in the same range of molar mass (Figure 7), are due to the copolymer branching, which is the only structural difference between both. Ethyl and longer branches do not enter the crystal and reside preferentially in the melt that surround the crystallites. In a homogeneous melt above equilibrium, it is very unlikely that J

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Figure 14. Plot of heat flow against temperature for cooling from melt temperatures of 140, 150, and 180 °C and subsequent heating at 10 °C/min for hydrogenated polybutadienes ArgP3.6 (left) and P98 (right). The dashed lines are drawn at the equilibrium melting, calculated with eq 1. Exothermic peaks point down, endothermic up.

the effect of melt-memory on crystallization vanishes with increasing branching. A more quantitative account is found in the plots of peak crystallization temperature Tc, against the initial temperature of the melt (Tmelt) (Figure 15). As a

crystallizable sequences long enough to enter the crystal reside within a close proximity. These sequences must diffuse through the melt to the crystallite leaving in this path a melt topology of ties and interconnected entangled chains. Upon melting above Tom copo, diffusion of the sequences back to the homogeneous melt, which is fast in the homopolymer, is more restricted in the copolymer due to the accumulation of branches in the surrounding region. Restrictions to diffusion vanish when the chain length is shorter than the required entanglement length, and as observed, melt memory vanishes for copolymer chains Tom copo in order to determine Tonset (critical value of Tmelt for melt homogeneity). The effect of melt memory on increasing Tc is clearly seen in copolymers with 2.2 mol % and 3.6 mol %, it is almost negligible for 4.14 mol % and vanishes for the copolymer with 5.68 mol %. The difference Tonset − Tom copo gives a means to quantify the strength of melt memory. The larger this difference is, the higher the temperature above Tom copo where heterogeneity from sequence partitioning still remains. This difference is plotted in Figure 16 for increasing branching content. We find that as the content of branches increases the difference becomes smaller, i.e., the melt becomes homogeneous at temperatures closer to the equilibrium value, or closer to thermodynamic expectations. The rational is that as the branching content increases and crystallizable sequences become shorter and more depleted, copolymers develop lower degrees of crystallinity and thinner crystallites, as is well documented.5−7,35−37,47 Although the K

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barrier for nucleation in a subsequent cooling, and hence, increases crystallization rate. The onset of melt memory (Tonset), or highest temperature of the melt above Tom copo that retains memory of the prior copolymer sequence partitioning, is molar mass and branching content dependent. For a fixed 2.2 mol % branches, Tonset increases linearly with the logarithm of molar mass and reaches the equilibrium value at a molar mass of 1300 g/mol. This molar mass corresponds to the critical entanglement molecular weight for polyethylene. Copolymers free of entanglements are free of melt memory above Tom copo. The strength of copolymer melt memory, or range of melt temperatures above Tom copo where the melt retains at least partial sequence partitioning, decreases with increasing branching content and vanishes at 4.53 mol % branches. We attribute this dependence to a much reduced crystallinity of highly branched copolymers and the formation of thinner crystallites. As crystallinity decreases, the amorphous region becomes less constrained allowing crystalline sequences to diffuse more readily back to the homogeneous melt state. The observed correlation between copolymer crystallinity and melt topology opens the path to strategies for controlling the state of copolymer melts in ways of technological relevance for melt processing of LLDPEs and other types of branched polyolefins.

Figure 16. Plot of difference between onset temperature for melt memory and equilibrium melting temperature against ethyl branching content.

restrictions to diffuse crystallizable sequences to a nucleus site are even more prevalent in copolymers melts with a lower number of suitable sequences, the reduced number of crystallites formed restricts topological constraints in the intercrystalline regions. Thus, diffusion to the randomized melt is relatively fast upon melting thin crystallites of highly branched copolymers. The memory strength or Tonset − Tom copo value is linearly proportional to the copolymer branching content, as shown in Figure 16, and extrapolates for Tonset − Tom copo = 0 to a critical branching content of 4.53 mol %. Hence, random ethylene 1-butene copolymers with >4.53 mol % branches are free of strong melt-memory, they crystallize from homogeneous melts at any Tmelt > Tom copo.

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ASSOCIATED CONTENT

S Supporting Information *

Tmelt vs Tc for ethylene − 1-hexene (1.87 mol %), data for original copolymer and after extraction, plot of heat flow rate vs temperature for P108 cycling from 40 to 150 °C eight times, and full scale WAXD and SAXS patterns. This material is available free of charge via the Internet at http://pubs.acs.org.

4. CONCLUSIONS Random ethylene copolymers display a strong memory of their prior crystallization even at temperatures above the equilibrium melting point (Tom copo). Such memory accelerates a subsequent crystallization as observed either by higher crystallization peaks on a dynamic cooling from the melt, or by shorter half times under isothermal crystallization. The memory effect above Tom copo is unique to random copolymers, and is absent in linear polyethylene fractions. By detailed experimental studies we attribute this memory effect to seeds made of molten clusters of ethylene sequences with little or no branching that remain as a memory of the crystallizable sequence length partitioning in the evolution of the initial crystallites. Morphological and structural data by POM and X-ray diffraction indicate that the seeds lack any crystalline nature and may have little, if any, structural order. We found that the strength of melt memory (number of seeds that remain at Tmelt > Tom copo) is molar mass dependent and requires a relatively high degree of the initial crystallinity of the copolymer. These features support the dynamic nature of the memory effect. We conclude that the origin of the self-seeds comes from the need to select suitable crystallizable sequences and transport them through the entangled melt to a crystal site. This process leaves an amorphous chain topology constrained by knots, loops, ties and other entanglements, such that upon further melting, diffusion of the partitioned sequences back to the homogeneous randomized melt state is a very slow process, hindered by this topology. For many copolymers diffusion to the randomized melt requires temperatures well above their equilibrium melting. The memory of sequence partitioning in copolymer melts above Tom copo effectively lowers the free energy

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AUTHOR INFORMATION

Corresponding Author

*E-mail: (R.G.A.) [email protected]. Present Address ‡

ExxonMobil. Research and Engineering Company, Clinton NJ 08801 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge initial discussion of our work with B. Goderis of the Department of Chemistry, Molecular and Nanomaterials, Catholic University of Leuven (K.U. Leuven), Belgium. Some experimental work was carried out by undergraduate students P. Esteso, and K. Thompson. Funding of this work by the National Science Foundation (DMR1105129) and by ExxonMobil Co. is gratefully acknowledged.

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