Roughness of growth faces of polymer crystals: Evidence from morphology and implications for growth mechanisms and types of folding

It is proposed that the growth faces of lamellar polymer crystals can have an equilibrium roughness (or crenellation). This can explain why some polymer crystals show no evidence of faceting. Support for this idea comes from the extensive theoretical developments on the nature of crystal surfaces. The characteristic habits of polyethylene are analysed in terms of a roughness which, on the {110} faces, increases progressively over a temperature range of about 100°C. At a temperature near 110°C the roughness becomes sufficient for there to be no free energy penalty for arbitrary crystal shapes (e.g. rounded) compared with one bounded by {110} faces. Above this temperature of crystallization most of the habits which are observed are leaf-shaped, with an apex along 〈010〉. Below 110°C{110} facets (or microfacets) are normally seen. There is no positive evidence that faces approximately parallel to (100) planes, observed for crystallization temperatures in the range 80°–110°C, are ever other than rough. The relative rates of growth on the {110} compared with the (100) increase with temperature, since {110} faces predominate at low but not high temperatures of crystallization. These changes are attributed to the increase in roughness with temperature on the {110} faces. The existence of surface disorder (roughness) requires that the binding energy between units in the crystal is comparable with KT. Hence this unit is probably several monomer units of polyethylene (rather than, for example, a complete stem which contains a hundred or more monomers). There is therefore a surface lattice on the growth faces with twenty or more units in the direction perpendicular to the lamellae. Monte Carlo calculations are cited for lattices of 20 by 50 units. These show that a cooperative increase in surface roughness with temperature and a transition between faceted and non-faceted growth can be expected for lattices of such limited extents. No explicit allowance has been made as yet for the consequences of the units being linked into chains and, for that reason, not being able to arrive or leave the surface independently. It is noted that changes in the alkane lattice with temperature indicate a possible evolution in binding energy, and in mobility, and hence may influence surface roughness. Theories of crystallization in polymers have normally assumed a growth surface which is molecularly smooth in equilibrium, and have emphasised nucleation events. Since this paper shows that the equilibrium structure may often be rough, it may be necessary to re-examine the basis of these theories. A brief review is included of the experimental evidence for surface nucleation events: nucleation may be a more important barrier at low temperatures than at high. The type of folding will be influenced by equilibrium roughness just as it will be by kinetic roughness, and some comparisons are made with neutron scattering results on this topic. The degree of adjacent folding is higher in the faceted regime as expected. Brief comments are made on the applicability of this idea to polymers other than polyethylene.     

A nucleation mechanism leading to stacking of lamellar crystals in polymer thin films

Systematic studies based on well‐controllable model systems aim at understanding how crystallization from a melt or solution of randomly coiled polymers leads to the formation of mono‐lamellar crystals. However, besides mono‐lamellar crystals also various other morphologically simple but yet not well understood structures are found. In particular, stacks of correlated lamellar crystals have been observed since the early days of the study of polymer crystallization. Here, we demonstrate that a recently proposed mechanism of self‐induced nucleation within lamellar crystals provides a possibility to explain how in such stacks lamellar crystals can be correlated. Examining various polymer systems, we show that the probability for generating self‐induced nuclei depends on the morphology of an initiating dendritic basal lamellar crystal. In addition, we provide evidence that this self‐induced nucleation mechanism, together with a high rate of transport of molten polymer to the fold surface, may allow the formation of polymer crystals with similar size in all three dimensions, containing a large number of superposed correlated lamellae. © 2019 The Authors. Polymer International published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry.     

A mechanistic growth model for inorganic crystals: Solid‐state interactions

Growth shapes of inorganic crystalline solids govern material properties such as catalytic activity and selectivity, solar cell efficiency, and so forth. A systematic understanding of the crystal growth process and the solid‐state interactions within inorganic crystals should help to engineer crystal shapes. A general model that identifies periodic bond chains in inorganic crystals while accounting for the long‐range electrostatic interactions is presented. The variation in the electronic structure and the partial charges of growth units on the inorganic crystal surfaces has been captured using the bond valence model. The electrostatic interaction energies in the kink sites of inorganic crystals were calculated using a space partitioning method that is computationally efficient. This model provides a quantitative explanation for the asymmetric growth spirals formed on the surface of calcite. This methodology for studying solid‐state interactions can be used with a mechanistic growth model to predict the morphology of a wide variety of inorganic crystals. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3707–3719, 2014     

A mechanistic growth model for inorganic crystals: Growth mechanism

Inorganic crystals grown from solution find wide application. A mechanistic growth model based on the spiral growth mechanism that operates at low supersaturation on inorganic crystal surfaces is presented. The long‐range electrostatic interactions on inorganic crystal surfaces are captured by methods developed in our previous article (Dandekar and Doherty, AIChE J., in press). The interactions of kink site growth units with the solvent molecules partially determine the growth kinetics. Relevant experimental parameters are systematically accounted for in the expression for the kink incorporation rate along step edges on the crystal surfaces. The growth model accurately predicts the asymmetric growth spirals on the surface of calcite crystals. The effect of supersaturation and ionic activity ratio on the step velocities of the acute and obtuse spiral edges is also correctly captured. This model can be used to predict the shapes of solution grown inorganic crystals and to engineer the growth process to design inorganic solids with functionally desirable shapes. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3720–3731, 2014     

Decrease in the isotropization temperature and enthalpy of main-chain polymer smectic liquid crystals as a result of the inclusion of chain ends

Main-chain liquid crystal (LC) PB-10-x_n polyesters with number average degrees of polymerization (x_n) ranging from 12 to 50 were prepared. The PB-10 polyesters formed smectic I (SmI) LCs that consisted of 40-nm-thick lamellae stacked along the polymer chain direction. Although the LC lamella thickness (d_LC) was largely independent of x_n, the isotropization temperature (T_i) of the SmI LCs decreased by 12 °C with decreasing x_n. Such a decrease in T_i can be explained by the inclusion of polymer chain ends in the LC lamellae, which resulted in the low isotropization enthalpy and curved LC lamellae, as observed in PB-10-12.     

Regime III crystallization in melt-crystallized polymers: The variable cluster model of chain folding

The kinetic nucleation theory of chain folding, including the effects of reptation, is extended to predict the increase in crystal growth rate G that is implied by measurements on PE and POM at moderately large undercoolings. The increased growth rate denotes the rather abrupt transition from Regime II where $\text{G∝i}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ to Regime III where G∝ i (i = surface nucleation rate). The distance between the niches on the growth front in Regime II diminishes rapidly with falling crystallization temperature T_x, and approaches the molecular width at a specified undercooling where Regime III begins. In PE, the Regime I → Regime II transition occurs at $\text{ΔT ? 16°}\text{C}$, and the Regime II → Regime III transition is predicted to occur at ΔT ～ 23°C (ΔT based on T°_m(∞) = 145°C). Growth rate data on PE and POM crystallized from the melt suggest conformity with the theoretical predictions. The implications of Regime III crystallization to chain morphology are discussed. The kinetic theory, which predicts narrowly spaced niches on the growth front, taken together with the restrictions on the degree of non-adjacent re-entry imposed by the ‘Gambler's Ruin’ treatment, leads directly to the ‘variable cluster’ model as the relevant morphology in Regime III. Here runs of adjacently chain-folded stems of varying size (averaging about three or so stems) are laid down interspersed with non-adjacent re-entries, leading to a lamellar surface that is about two-thirds ‘regular’ or ‘tight’ folds, most or all of these representing strictly adjacent re-entries. The steady-state reptation process operative in Regimes I and II in PE is impaired at temperatures just below the inception of Regime III, and it is suggested that at lower temperatures the ‘slack’ portions of the chains engage in forming the small clusters of adjacent stems. The variable cluster model leads in a natural way to the amorphous component found in quench-crystallized polymers, and is consistent with neutron scattering data on quench-crystallized PEH-PED.     

Lamellar thickness transition of melt-crystallized polybuten-1 tetragonal phase: configurational change in chain folding directions

Lamellar crystal thickness l _c of isotactic polybutene-1 (it-PB1) have been investigated for crystallization in the melt over a wide range of crystallization temperature T from 40°C to 90°C by small angle X-ray scattering experiments and density measurements. The crystal thickness l _c demonstrates two linear dependences on inverse supercooling and a transition from one dependence to the other has been observed around T = 65°C. Each of the two dependences obeys the nucleation theory in the high and low supercooling ranges, respectively. Chain folding free energy q determined from the low supercooling range is larger than that determined from the high supercooling range. Possible mechanisms for the transition are discussed taking account of entropy of chain folding directions.     

Kinetics of the ordered phase growth at the phase transition from the isotropic to cholesteric state in a main chain liquid crystalline polymer

The optical microscope investigation of the isotropic- ordered phase transition in a main chain LC polymer and subsequent statistical treatment of the microscopic images allowed recognition of three regimes of the ordering: non-stationary regime which is characterized by increasing rate of the droplets growth; stationary regime; within it the ordered phase grows with a constant rate, and the coalescence regime for which the rate of the droplets growth deceases. Linear interpolation of the time dependence of the mean droplets diameter in log–log scales allowed conclude that within the stationary regime of the cholesteric phase growth, log <d> is a linear function of log t, whereas within the coalescence regime, <d> is proportional to t ^1/2 as expected from the theory of phase separation binary liquids. The time of beginning and duration of both growth regimes of the phase transition for the main chain LC polymer exceed those for low molecular weight LC compounds. This is caused by lower mobility of macromolecules in comparison with that for low molecular weight LC compounds.     

A method to determine the relative value of the barriers of carbon chain growth and termination in Fischer-Tropsch synthesis: elementary reaction kinetics for chain growth

A method is established, by which the difference of the reaction activation barriers of carbon chain growth and termination in Fischer-Tropsch (FT) synthesis can be determined from experiments. A FT synthesis is carried out on Fe/Zn catalyst. We apply the method to analyze the experimental result and obtain the difference of reaction activation barriers of carbon chain growth and termination of -olefins on the catalyst.     

A transport model for the absorption of mixed solvents in glassy polymer membrane

Transport model for mixed solvents in glassy polymer membrane is rare in literature. In our previous work, a new experimental method has been developed and absorption kinetic curves for two mixed solvent systems (ethanol/1,2-dichloroethane and ethanol/ethyl acetate) in polyurethane (PU) membrane at have been measured. In this work, based on Liu et al.'s transport model for single solvent/polymer membrane system, a transport model for the absorption of mixed solvents in glassy polymer membrane is established. Three model parameters in this model can be obtained by correlating the experimental data of the corresponding single solvent/polymer membrane systems; the other three should be determined by correlating the experimental data of mixed solvents/polymer membrane system. The effect of model parameters on diffusion is studied by numerical simulation. The correlated results agree well with the experimental absorption curves. The model has the ability to predict the transport phenomena of mixed solvents in polymer membrane.     

An electrochromic diquat-quaterthiophene alternating copolymer: A polythiophene with a viologen-like moiety in the main chain

A bis-bithiophenyl derivative of diquat, 3,10-bis(2,2′-bithiophene-5-yl)-6,7-dihydropyrido[1,2-a:2′,1′-c]pyrazinodiynium hexafluorophosphate (bt2dq), was synthesized by quaternization of 5,5″-bis(2,2′-bithiophene-5-yl)-2,2′-bipyridine with 1,2-dibromoethane. Its UV–vis absorption spectrum is explained by TD-DFT calculations. It shows the electrochemical properties characteristic for bipyridinium salts (viologens and diquats) and can be electropolymerized to form a conjugated polymer composed of alternating quaterthiophene and diquat blocks. The polymer has been characterised by cyclic voltammetry and UV–vis spectroelectrochemistry: it can be reversibly oxidized, with spectral signs of p-doping of the oligothiophene blocks, and reversibly reduced, with formation of viologen-like cation radicals, which dimerize or form π-stacks.     

Theory of the substrate length in polymer crystallization: Surface roughening as an inhibitor for substrate completion

A model is proposed for the physical origin of the substrate length L that appears in the customary treatment of the regime I→II growth rate transitions which occur in certain polymers during crystallization from the melt. (A previous analysis of growth rate measurements showed that L ≈ 0.77 μm at the I→II transition in polyethylene). L is treated as a ‘persistence length’ between defects that have the capacity to inhibit substrate completion. The defects are pictured as resembling the Greek letter Ω (omega) in their most extended state; in their normal state they are represented as hemispherical or disc-like amorphous patches that are pinned onto the substrate. The omega defect can form on the substrate by drawing in a portion of one of the cilia, loose loops, or interlamellar links that are characteristic of the ‘variable cluster’ representation of the molecular morphology of lamellar semicrystalline polymers. The formulation relates L to the equilibrium free energy of formation of the omega defect, which is viewed as being principally entropic. Thus we derive $\text{L∝(}\text{stem width}\text{) ×}\text{exp}\text{(?}\text{ΔS}\text{R}\text{)}$. From the known value of L for polyethylene, it is determined that the experimental entropy of formation of the defect is ΔS_expt. = ? 12.6±1.5 cal mole^?1 deg^?1. This is justified on basic grounds by first applying nucleation theory to estimate the number of chain units $\text{n}{}_{\mathrm{\Omega }}$ in the defect of critical size. Then from partition functions for once- and twice-pinned polymer chains on a surface, which gives $\text{ΔS = ?fR}\text{ln}\text{n}{}_{\mathrm{\Omega }}$ with f～ 2.0 to 2.5 depending on defect shape, one arrives at a theoretical estimate of ΔS for the omega defect in polyethylene that is in good agreement with the experimental value. This indicates that the omega defect model for L is reasonable on energetic grounds. It is shown further that the model is consistent in a number of respects with what is known about the I→II transition and L. Criteria for the occurrence of I→II transitions are presented, and the range of validity of the theory is discussed. It is noted that the I→II transition may be diffuse or absent in many cases, either because the equilibrium distribution is not attained or because the lifetime of the defects is too short in comparison with the residence time. Thus in many polymers, regime I may be missing so that regime II (with its locally rough growth front) will persist up to quite high temperatures, i.e., up to the practical limit of slow growth.     

A functional monomer to synthesize sulfonated poly(ether ether ketone) with sulfonic acid group in the pendant side chain

A functional monomer which enables the synthesis of sulfonated poly(ether ether ketone) (SPEEK) polymer with the sulfonic acid group in the pendant side chain has been successfully developed. 1, 3‐propane sultone was used as the model compound to introduce the side chain moiety in the monomer. The above sulfonated monomer (0.50 mol) along with 4, 4′‐difluorobenzophenone (0.50 mol) were reacted with bisphenol‐A (1.0 mol) to obtain the SPEEK polymer with the sulfonic acid group in the pendant side chain. All the intermediates and the SPEEK polymer were characterized using ¹H‐NMR. The SPEEK polymer is expected to have improved mechanical and electrical properties for PEM fuel cell application. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2012     

Kinetic analysis for crystal growth rate of NH4Cl in the NaCl‐MgCl2‐H2O system with a thermodynamic approach

A growth kinetic model has been developed from a rigorous thermodynamic perspective to describe the crystal growth rates of NH₄Cl on the basis of the difference of chemical potentials of NH₄Cl at solid–liquid interface in aqueous NH₄Cl, NH₄Cl‐NaCl, and NH₄Cl‐MgCl₂ solutions. The solid–liquid equilibrium and activity coefficient of NH₄Cl are calculated by the newly developed accurate Pitzer model with aid of Aspen Plus? platform. The predictions of the resulting model are in good agreement with the experimental data published in literature and determined in this work at 283.15–333.15 K within the supersaturation up to 0.1. The kinetic model was subsequently used to analyze the effect of several operation variables, including temperature (283.15–333.15 K), supersaturation (up to 0.1), and NaCl or MgCl₂ concentration (0～2.5 mol kg^?1), on the crystal growth rate of NH₄Cl. The crystal growth rate of NH₄Cl, with activation energy of 39 kJ mol^?1, is strongly temperature‐dependent and increases with increasing temperature in the three systems investigated. The advantage of MgCl₂ over NaCl on the recovery of NH₄Cl is theoretically and experimentally illustrated from the thermodynamic and kinetic perspectives with the aid of the established model. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

Dynamic pH model for autotrophic growth of microalgae in photobioreactor: A tool for monitoring and control purposes

A dynamic model for the photoautotrophic growth of microalgae in photobioreactor that describes the main variables of the system and allows the precise prediction of the pH in the culture was proposed and validated. The dynamic behavior of the biological system was expressed through a multistate model in continuous‐time formulation, based on mass‐balance equations and local photosynthetic responses of the anisotropic medium, further associated with a set of algebraic equations that describes the thermodynamic properties of the ammonia—carbon dioxide—water ternary solute system. The global photoautotrophic growth model was validated on experimental data acquired from a torus reactor inoculated with Chlamydomonas reinhardtii cells. The model response was studied in simulation for all identified input variables (dilution rate, incident light intensity, temperature, and flow rates of input gases). © 2013 American Institute of Chemical Engineers AIChE J 60: 585–599, 2014     

Heat treatment of cream: A model of the butter texture response in relation with triglyceride composition

The effects of thermal treatments on butter texture are known and have been used since 1935 on an industrial scale, but without fundamental knowledge. Butter composition influences firmness, as observed through seasonal and regional variations. Experiments were carried out at 15°C by using a cone penetrometer and an industrial testing machine. A significant correlation between heat treatment efficiency and some prevalent triglycerides and fatty acids on butter firmness was outlined. Three fatty acids (myristic, oleic, palmitic) and four major groups of triglycerides mainly affected the firmness, sometimes leading to an inversion of the thermal effect, according to individual sample composition. A crystallographic and thermodynamic model based on triglycerides properties was developed.     

A macroscopic model for species transport during in vitro tissue growth obtained by the volume averaging method

In this work, we study the possibility of deriving a macroscopic model to describe reaction and transport by diffusion and convection of two species within a porous medium as encountered during in vitro tissue growth. The starting point is a boundary value problem of diffusion-advection and reaction in a three-phase system, the two species being identified as the nutrient for cell growth and the metabolic product within the framework of tissue culture. The method of volume averaging is applied to the set of microscopic equations. Under the local mass equilibrium assumption and a series of constraints on the parameters of the system that are identified, one obtains a one-equation macroscopic model corresponding to a dispersion-reaction equation. Associated closure problems allowing the computation of effective coefficients that appear in this macroscopic model are provided.     

Optimal design and operations of supply chain networks for water management in shale gas production: MILFP model and algorithms for the water‐energy nexus

The optimal design and operations of water supply chain networks for shale gas production is addressed. A mixed‐integer linear fractional programming (MILFP) model is developed with the objective to maximize profit per unit freshwater consumption, such that both economic performance and water‐use efficiency are optimized. The model simultaneously accounts for the design and operational decisions for freshwater source selection, multiple transportation modes, and water management options. Water management options include disposal, commercial centralized wastewater treatment, and onsite treatment (filtration, lime softening, thermal distillation). To globally optimize the resulting MILFP problem efficiently, three tailored solution algorithms are presented: a parametric approach, a reformulation‐linearization method, and a novel Branch‐and‐Bound and Charnes–Cooper transformation method. The proposed models and algorithms are illustrated through two case studies based on Marcellus shale play, in which onsite treatment shows its superiority in improving freshwater conservancy, maintaining a stable water flow, and reducing transportation burden. © 2014 American Institute of Chemical Engineers AIChE J, 61: 1184–1208, 2015     

Simulating the variability and scale effect for slow crack growth in Hi-Nicalon SiC-based tows: A parametric study

The delayed failure time of SiC-based multifilament tows under static fatigue condition is as broadly scattered as for individual filaments, despite it is commonly used as strong scatter reduction. Moreover, the stress exponent (n) is hierarchy-dependent as revealed by a Monte-Carlo algorithm: decreasing from filament (micro) scale to tow (meso) scale. This is demonstrated to originate from the mismatch between the stress applied to the critical filament (affecting the growth kinetics), variable because of fiber misalignment in the tow, and its tow-averaged value used for endurance diagram construction. In the context of this algorithm, it is shown that n would evolve with fiber parallelism, tow stress range or the critical filament rank whereas filament strength distribution plays a secondary role. The tow structure shall therefore be considered as a major parameter for composite part design aiming long service life.