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- Herausgeber: Wiley-VCH Verlag GmbH & Co. KGaA
- Kategorie: Fachliteratur
- Sprache: Englisch
Introducing a unique, modular approach to modeling polymerization reactions, this useful book will enable practitioners - chemists and engineers alike - to set up and structure their own models for simulation software like Predici®, C++, MatLab® or others. The generic modules are exemplified for concrete situations for various reactor types and reaction mechanisms and allow readers to quickly find their own point of interest - a highly useful information source for polymer engineers and researchers in industry and academia.
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Seitenzahl: 445
Veröffentlichungsjahr: 2018
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Table of Contents
Cover
Title Page
Copyright
Preface
Chapter 1: Introduction
1.1 Special Features of Polymers
1.2 Structures in Polymers and Their Influence on Processing and Application Properties
1.3 Some Analytical Methods for Model Validation
1.4 Description of Polymer Properties
Chapter 2: Polymer Reactions
2.1 Module Concept
2.2 Rate Coefficients in Polymerization Reactions
2.3 Building Macromolecules
2.4 Only Chain-Forming Reactions Take Place, Step-Growth Polymerization
2.5 Chain-Growth Polymerization – Initiation Required
2.6 Copolymerization
2.7 Nonlinear Polymerization
2.8 List of Modules
Chapter 3: Reactors for Polymerization Processes
3.1 Introduction
3.2 Well-Mixed (Ideal) Batch Reactor (BR)
3.3 Semi-Batch Reactor (Semi-BR)
3.4 The Continuous Stirred Tank Reactor (CSTR)
3.5 Tubular Reactors
3.6 Nonideal Reactor Models with Partial Backmixing
3.7 Comparison of Reactors
Chapter 4: Phases and Phase Transitions
4.1 Treatment of Volumes and Concentrations
4.2 Phase Transfer Modules
4.3 Multiphase Polymerization Systems
Chapter 5: Numerical Methods
5.1 Introduction
5.2 Ordinary Differential Equations
5.3 Countable Systems of Ordinary Differential Equations – CODEs
5.4 Estimating the Numerical Error
5.5 Monte Carlo Methods
5.6 The Modeling Cycle: Dealing with Different Errors
Chapter 6: Parameter Estimation
6.1 Introduction: Forward and Inverse Problems
6.2 General Theory
6.3 Correlated Parameters
6.4 Example: Parameter Dependencies and Condition
Chapter 7: Styrene Butadiene Copolymers
7.1 Model Description
7.2 Components of the Model
7.3 Reaction Modules
7.4 Exemplary Simulations
7.5 Exemplification of the Modeling Cycle for the Styrene–Butadiene Example
References
Appendix
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Introduction
Figure 1.1 Possible representations of distributions: (a) discrete, differential frequency, or number distribution of degree of polymerization, (b) continuous differential frequency or number distribution of molar mass, (c) discrete, cumulative distribution of degree of polymerization, (d) continuous, cumulative distribution of molar mass.
Figure 1.2 Schematic entanglement of bulk polymers.
Figure 1.3 Shear viscosity curves for polyethylene with different weight average molar mass.
Figure 1.4 Shear viscosity curves of polyethylene grades with different molar mass distributions (normalized to same zero-shear viscosity).
Figure 1.5 Schematic dependence of melt viscosity on molar mass for different architectures.
Figure 1.6 Typical glass transition temperature behavior of statistical and block copolymers.
Figure 1.7 -centered triads and pentads.
Figure 1.8 Schematic topology of a branched macromolecule with backbone chain, branches, and branches on branches. The overall chain length of the molecule is given by the total number of monomers in all branches.
Figure 1.9 Schematic monomer distribution in an example terpolymer chain.
Figure 1.10 Two possible paths for the number of molecules disappearing with a certain rate. In the first picture, the status of the system is 7 at time 0, then 6 at time 1, then still 6 at times 2 and 3 (7–6–6–6). This means that in only one of the three time intervals the reactions really happened. For the second path, the reaction takes place in the second and third time step, leading to status 7–7–6–5 at times 0–1–2–3. The difference is simply given by the underlying stochastic process. In order to get a reasonable average, one has to repeat this very often.
Figure 1.11 Time evolution of PDF of for until the average reaches . The PDF presents the probability that there are molecules (-axis) present in the system at time . In the beginning we have , then the PDF broadens and moves from right to left.
Figure 1.12 Commutative diagram for CME and RKE. Note that the upper path usually cannot be performed with high accuracy, since the CME cannot be solved directly.
Figure 1.13 Diagram for CME and RKE including methods.
Chapter 2: Polymer Reactions
Figure 2.1 Different representations of the chain-length distribution resulting from condensation of according to (2.37) after 200 s. Conditions: mol/l, l/mol/s.
Figure 2.2 Variation in time of number and weight average, dispersity index, and moment resulting from condensation of according to Equation (2.37). Conditions: mol/l, l/mol/s.
Figure 2.3 Frequency and mass distribution of and for a reaction of and after 200 s. Conditions: mol/l, mol/l, l/mol/s.
Figure 2.4 Development in time of overall number and weight average molecular weight and of chain concentration of and for a reaction of and . Conditions as in Figure 2.3.
Figure 2.5 Exemplary distributions resulting from polymerization for an identical number of end groups.
Figure 2.6 Exemplary distributions resulting from polymerization for .
Figure 2.7 Principal development in time of the number averages of the three distributions for the stoichiometric and non-stoichiometric case .
Figure 2.8 Randomization of a bimodal distribution to a Schulz–Flory distribution when only the interchange reaction (2.108) by intermolecular end group attack takes place.
Figure 2.9 Comparison of Gold and Poisson distribution for .
Figure 2.11 Evolution of the distribution in a living system with propagation and depropagation.
Figure 2.10 Reduced rates of propagation, depropagation, and overall rate of polymerization as a function of temperature.
Figure 2.12 Frequency distribution of living () and dead () chains for various according to Equation (2.194) assuming instantaneous initiation and full monomer conversion.
Figure 2.13 Broadening of the distribution by deactivation reaction during polymerization with instantaneous initiation at different monomer conversions.
Figure 2.14 Radical concentration during the first phase of radical polymerization until the steady state is reached.
Figure 2.15 Typical periodic profile of radical concentration in pulsed-laser polymerization.
Figure 2.21 Development of calculated from all monomer-consuming reactions for different compared to computed according to the QSSA and LCH assumption for .
Figure 2.16 Copolymerization diagram giving the instantaneous composition of the polymer as a function of the composition of the monomer mixture for some example values of and .
Figure 2.17 Average chain lengths of sequences of monomer and as a function of for , .
Figure 2.18 Bonding frequencies as a function of mole fraction for , .
Figure 2.19 Run number for various combinations of reactivity ratios as a function of mole fraction of .
Chapter 3: Reactors for Polymerization Processes
Figure 3.1 Change of MMD with number of CSTRs according to the cell model (see Section 3.4.2) for a living polymerization with .
Figure 3.2 Runaway in living styrene polymerization.
Figure 3.4 Schematic representation of a CSTR.
Figure 3.3 Instantaneous and integral values of and composition .
Figure 3.5 Exemplary dynamic simulation of the development of for a living polymerization in a CSTR for two different start-up procedures.
Figure 3.6 Scheme for a plug-flow reactor with partial recycling.
Figure 3.7 Comparison of cumulative residence distribution function for various reactor models.
Chapter 4: Phases and Phase Transitions
Figure 4.1 Scheme of possible phases in a reactor.
Figure 4.2 General scheme for the condition at the boundary between two phases, I and II. , , concentration of component in phase I resp. II. , , concentration of component at the boundary layer of phase I resp. II. , molar flow of component from phase II to phase I.
Figure 4.3 Equilibrium pressure according to Antoine equation, actual pressure and volume of the liquid phase during heating.
Figure 4.4 Mole fraction of the more volatile compound of a binary mixture in the liquid phase during heating and cooling for different values.
Figure 4.5 Vapor pressure reduction above a polymer solution as a function of polymer content for different values of .
Figure 4.6 Onset of phase separation when feeding a polymer solution to a poor Solvent.
Figure 4.7 Monomer concentration in the gas and liquid phases and mass of polymer during batch and fed-batch polymerization with a constant feed of gaseous monomer.
Figure 4.8 Typical phenomena in emulsion polymerization.
Figure 4.9 The three stages of emulsion polymerization. , monomer concentration in latex particles, , number of particles, , surface tension, , rate of polymerization.
Figure 4.10 The problem of compartmentalization. (a) The radical chains are distributed among the particles, and termination is only possible in two of the particles. (b) The cartoon shows all radicals packed into one big particle; thus, they may all react with each other.
Figure 4.11 Particle-radical distribution for several values of entry rate , values listed in Table 4.1.
Chapter 5: Numerical Methods
Figure 5.1 Exact solution (−) and explicit Euler steps of size for ODE .
Figure 5.2 Explicit Euler scheme applied to (5.13) with and .
Figure 5.3 Implicit Euler scheme applied to (5.13) with and .
Figure 5.4 Real and estimated error of the implicit Euler method with . The estimated error is obtained by comparison to the solution using the half . Apparently, the real error is very large and the estimate is not a good measure, since it is based on a solution that is itself not very accurate.
Figure 5.5 Real and estimated error of the implicit Euler method for the test example using a relative error with scaling threshold. Left: . Right: .
Figure 5.6 MEC method applied to test example. The dashed line describes the first-order approximation, whereas the second-order correction cannot graphically be distinguished from the exact solution.
Figure 5.7 Adaptive step sizes taken from a simulation of a radical polymerization with intermediate feed points using the MEC method.
Figure 5.8 (Normalized)Poisson distribution generated by (5.41) with upper chain-length index .
Figure 5.9 Two ways to solve countable system (CODEs). The upper path (method of time layers) and the lower direction (method of lines) use different principles to discretize time- and chain-length domain.
Figure 5.10 Moving front in a transport equation PDE. Higher resolution is required at the front.
Figure 5.11 Two time cuts of a chain-length distribution in a living polymerization. The grids are computed adaptively by a Galerkin method (Predici).
Figure 5.12 Feed may change the shape of a distribution dramatically, it may get even more complex or just simpler and the chain-length range may also vary in a different way. On the left-hand side we see the result of a short additional initiator pulse, on the right-hand side the effect of a constant low initiator addition in a living polymerization.
Figure 5.13 Two ways to compute moments and mean values of a chain-length distribution. The upper direction approximates by some algorithm and then uses a rigorous summation, the lower path averages the basic equations first and then performs the time integration of the resulting ODE system.
Figure 5.14 The general model cycle applied to a typical situation.
Chapter 6: Parameter Estimation
Figure 6.1 Plots [124] of three typical types of objective functions in parameter estimation. (a) Unique global and thus local minimum. (b) Multiple local minima with one global minimum. (c)“Flat well” case with a region of connected local minima.
Figure 6.2 Exemplification of a very well-conditioned minimization problem (condition ). The condition number can roughly be estimated by the ratio of the axes of the red ellipsoid. A thin ellipsoid, as indicated in light yellow, would mean a condition of about .
Figure 6.3 Three-dimensional plot of the objective function of a simple fit problem with condition from radical polymerization with three parameters that are all sensitive, all possible pairings of the parameters are uncorrelated, but the set of all three is numerically correlated. The blue color marks small values, and apparently there is a whole region where parameter combinations may be (nearly) equivalent. The white line shows a typical path of iteration steps given by a Gauss–Newton method with reduced directions. A standard damping method would stop somewhere outside due to huge damping factors.
Figure 6.4 Time-dependent graphics for initiator and monomer , the efficiency , the end group concentration, and the mean values and of the dead polymer for the model in (6.42) used for generation of “experimental” data.
Figure 6.5 Result of parameter estimation based on monomer data and fit of efficiency.
Figure 6.6 Result of simultaneous parameter estimation for , , and based on monomer concentration, average molecular weight, and full molecular weight distribution. Estimated rate coefficients in line 7 of Table 6.5).
Figure 6.7 Simulation results from the “estimation” model with estimated rate coefficients in line 7 of Table 6.5 and recipe from Table 6.4 compared to the results with the “real” model.
Chapter 7: Styrene Butadiene Copolymers
Figure 7.1 Individual and overall conversion.
Figure 7.2 Overall number and weight average.
Figure 7.3 Frequency and GPC (on linear scale) representation of overall molar mass distribution at the end of the reaction.
Figure 7.4 Frequency distribution of individual populations.
Figure 7.5 Exemplary poly(butadienyllithium) (a) and poly(styryllithium) (b) chains from Monte Carlo simulation. Blue dots represent styrene, and red dots butadiene units.
Figure 7.6 Average sequence lengths asl-S of styrene and asl-Bu of butadiene and bonding frequencies.
Figure 7.7 Temperature, monomer concentrations, and fraction of active Li-organyls.
Figure 7.8 Change of average molar mass with time.
Figure 7.9 Individual and overall molar mass distribution in frequency representation at the end of the reaction.
Figure 7.10 Monte Carlo simulation of individual chains. Top, psli; bottom, psh; Blue, butadiene; Red, styrene.
Figure 7.11 Overall sequence length distribution for styrene and butadiene (frequency distribution normalized to area).
Figure 7.12 Molar fraction of butadiene in single chains .
List of Tables
Chapter 1: Introduction
Table 1.1 Heat of polymerization of some example monomers [3, 4]
Table 1.2 Methods for determining macromolecular structures
Chapter 2: Polymer Reactions
Table 2.1 Typical functional groups in step-growth polymerization
Table 2.2 Examples for initiation in chain-growth polymerization:
Table 2.3 Variation of with initiator concentration and ratio of rate coefficients
Chapter 3: Reactors for Polymerization Processes
Table 3.1 Comparison of different reactor types
Chapter 4: Phases and Phase Transitions
Table 4.1 Example results from the Smith–Ewart equation (4.62) for different entry rates
Chapter 6: Parameter Estimation
Table 6.1 Rate coefficients used in the “real” model
Table 6.2 Recipe for generation of “experimental” data
Table 6.3 “Experimental” data generated from the “real” model used for PE
Table 6.5 Summary of parameterestimation attempts ( = value fixed)
Table 6.4 Alternative recipe
Modeling and Simulation in Polymer Reaction Engineering
A Modular Approach
Klaus-Dieter Hungenberg andMichaelWulkow
Authors
Prof. Klaus-Dieter Hungenberg
Ortsstrasse 135
69488 Birkenau
Germany
Dr. Michael Wulkow
Harry-Wilters-Ring 27
26180 Rastede
Germany
Cover image was provided by Lisa Kulot
All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
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© 2018 Wiley-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN: 978-3-527-33818-4
ePDF ISBN: 978-3-527-68575-2
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Mobi ISBN: 978-3-527-68576-9
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Cover Design Grafik-Design Schulz, Fußgönheim, Germany
Preface
Synthetic polymers have an overwhelming importance in our world. Moreover, this does not just mean the economic importance but also the role synthetic polymers play to overcome the challenges and trends in our world. Polymers help meet the customer's key needs in transportation, energy, housing, health, and so on. To fulfill these challenges, polymers must be tailored to the specific needs in their final application. The crucial task is the combination of the correct chemistry and the best process in order to link the monomer units to obtain an appropriate microstructure of the polymer: composition, chain lengths, branching, and so on, as well as their respective distributions.
Mathematical models have, for a long time [1, 2], helped describe the interdependency between the formation of macromolecules and their resulting structure in a quantitative manner and thus may help the product developing chemist in the laboratory when designing a new polymer as well as the chemical engineer when designing a plant. Moreover, models may serve as a link between both.
Thus, the book addresses the interest of both chemists and engineers, those who are already advanced practitioners, and also students and those starting on the topic of polymerization process modeling.
However, with this book we do not want to supply the reader with ready-to-use models for various cases, but to enable him or her to set up own models suitable to solve specific problems. For this purpose, we follow a modular, unifying approach which differs from most of the work done in this field in the past.
For every polymerization mechanism, separate models under certain assumptions and restrictions have been developed and successfully used. So, models for step-growth polymerization (see Section 2.4) for the description of polyesters, polyamides, polyurethanes, and so on, are in many cases restricted to yield averages and moments of the molecular mass distribution (MWD) by assuming the validity of the most probable Schulz–Flory distribution for the computation of the MWD. For the large class of chain-growth polymerization (see Section 2.5), there exist different models for ionic polymerization, living polymerization, radical polymerization, transition-metal-catalyzed polymerization, and so forth. All of them need certain assumptions. Even the ratio of rate coefficients causes different ways to solve for the MWD. In the case of living anionic polymerization (see Section 2.5.1), this results in either a Poisson or the so-called Gold distribution depending on the ratio of initiation and propagation coefficient. In radical polymerization and copolymerization, a steady-state approximation for the active species is often assumed. Models for heterogeneous processes like emulsion, dispersion, or precipitation polymerization form another class of models that usually differ in the assumptions for mass transfer steps.
To overcome the limitations of these case-by-case models, we introduce a modular approach. By this, reaction schemes of typical polymer reactions will be designed by a combination of a set of elementary reaction steps [2] and the corresponding rate equations. These reaction schemes will be put into any kind of reactors and their compartments and phases, where the transport rates for mass and energy between these compartments are again described by rate modules.
In Chapter 1 – as a kind of appetizer – we address some special features of polymer structures and processes compared to low molecular compounds, and give some examples for the dependence of application properties on the molecular structure and some hints to analytical methods to get an insight into the molecular structure we want to describe with our models. Moreover, we introduce the principles of deterministic modeling using differential-algebraic equations and stochastic modeling using Monte Carlo methods.
Chapters 2 to 4 are devoted to model building using this modular approach. Here, model building means the translation of the expert knowledge of the chemist or engineer about the reaction mechanism, the kinetics and thermodynamics, and the reactor into mathematical equations to quantify this knowledge.
Chapter 2 describes the modules for the elementary reaction steps used to describe polymer reactions. Even though not necessary, in many cases we link the modules to the more conventional description to familiarize the reader with this concept. With some examples the link between kinetics and structure is demonstrated.
In Chapter 3, reactor modules are derived with special emphasis on heat and mass balances and residence time distribution, together with some insight into how structural properties of the polymers depend on the residence time distribution of the reactor.
The use of different phases and the transport between them is described in Chapter 4 together with some simple chemical engineering approaches for the phase transfer in general, and how these principles can be applied to multiphase polymer systems like suspension, precipitation, or emulsion polymerization.
After having set up the model, we need methods to solve the equations. These are described in Chapter 5. However, this chapter does not describe the various methods to solve ordinary or partial differential equations in all detail, but highlights some generally important aspects like convergence, stability, and error control, and tries to give some hints to detect errors. We also direct the users to deterministic (Section 5.3) and stochastical solutions (Section 5.5) of the special equations related to polymer kinetics.
Essential for the quality of the modeling results is the quality of the model parameters. So, in Chapter 6 the problem of parameter estimation is discussed especially in view of the ill-posed nature of the underlying inverse problem and the dependency of parameters. An illustrative example shows how the number and kind of measurements used for the estimation influences the quality of the parameters.
For the execution of a modeling project, we propose to follow a modeling cycle (Section 5.6) to capture possible numerical and modeling errors in the model.
A detailed example system in Chapter 7 demonstrates the various aspects and techniques of modeling polymerization reactions.
We hope that these techniques force the reader to set up his or her own models and simulations to solve specific problems, and so we explicitly abstain from a chapter on optimization.
Klaus-Dieter Hungenberg Birkenau, GermanyMichael Wulkow Rastede, Germany
2017
Chapter 1Introduction
1.1 Special Features of Polymers
Why write a book especially on modeling of polymerization reactions? To answer this question, it is best to compare the production of polymers with the production of low-molecular-mass compounds and to see what is special for polymers. For example, to produce acetic acid
several processes can be followed. Rather old ones, already known to the ancient Egyptians and Greeks, used the biotechnology way from grapes via ethanol to vinegar. Or more modern techniques like the oxidation of acetaldehyde by oxygen and Mn(OAc) as catalyst, the air oxidation of butane at high pressures or the Monsanto process, the addition of carbon monoxide to methanol catalyzed by rhodium complexes. Despite the variety of raw materials and processes, all end with acetic acid with the structure shown earlier. The various processes only differ in terms of raw material, conversion, yield, selectivity, concentration, and kind of impurities; and the engineering task is to optimize these quantities, the product being in all cases acetic acid with defined properties.
This is by far not the case with polymers. Let us have a look at a simple polymer like polyethylene. It can be produced by several different mechanisms – by radical polymerization to give low-density polyethylene (LDPE) with short- and long-chain branches, or catalyzed by various transition metal catalysts from chromium, titanium, and so on to high-density polyethylene (HDPE) or linear low-density polyethylene (LLDPE) by opening the double bond of ethylene and linking monomers to chains of length .
However, even for such a simple polymer like polyethylene, there exists a wide variety of grades that differ in properties like viscosity, crystallinity, transparency, gloss, and so on, which means that these grades have different molecular structures; so polyethylene is not just polyethylene. One obvious difference between two polyethylene molecules might be the number of monomeric units in the chain, the degree of polymerization. In contrast to biological macromolecules like enzymes, DNA, and RNA, which have well-defined structures despite being polymers, synthetic macromolecules (and also many of the biopolymers like cellulose, lignine, etc.) differ at least in length , and consist of an ensemble of chains with different chain lengths – they have a chain-length distribution. There might be more structures differentiating polymer chains from each other, and we come back to the kind of structural differences later in Section 1.2. These molecular structures determine the properties which the polymer will show during processing and in their final application. The molecular structure, however, depends strongly on the chemistry, the kinetics, and the process conditions.
Therefore, the challenge for chemists and engineers in the development and production of synthetic polymers is not only to optimize yield, conversion, and so on but also to produce the right molecular structure to meet the desired application and processing properties.
Here, suitable mathematical models linking reaction kinetics and process conditions to the resulting structure will be of great help to overcome this challenge. Especially if we consider that for polymers, in contrast to low-molecular-mass compounds where we have distillation, extraction, crystallization, and other purification methods, there does not exist any separation method – except on an analytical scale – to separate polymers with the desired structure from “bad” polymers. So, the polymerization process itself must yield the desired structure at once.
There is another important difference between processes for polymers and low molecular compounds. We have to consider that the chain length might be several hundreds or thousands or even higher. This means that the molecular mass (which is given by times molecular mass of the monomer unit) can be as high as several or even g/mol, so we have to consider processes where a low molecular compound, the monomer, with a low viscosity like water reacts to products with much higher viscosity, which may be up to Pa s. In the latter processes, however, we convert low-molecular-mass compounds to other low-molecular-mass chemicals, and viscosity will not change very much and will be Pa · s or even lower if we deal with processes in the liquid or gaseous state.
This dramatic change in viscosity has several implications. So, mixing becomes an issue. Poor mixing causes inhomogeneity with respect to concentration, and also to temperature. We have to keep in mind that polymerization reactions are usually rather exothermic reactions (see Table 1.1) with adiabatic temperature rising up to several hundred or thousand K. High viscosity will impede heat removal capacity tremendously and may cause hot spots in the reactor that may lead to side reactions or even runaways.
Table 1.1 Heat of polymerization of some example monomers [3, 4]
Monomer
General formula
(kJ/mol)
(K)
Ethylene, propylene, styrene, vinyl esters, acrylic acid and -esters, dienes
H
2
CCHX
67–105
300–2000
Isobutene,
-methyl styrene, methacrylic acid and esters
H
2
CCXY
33–59
140–400
-Caprolactame, 2-pyrrolidone (
-butyrolactame)
0–17
0–70
In addition, this high viscosity may affect the reaction kinetics itself, as in some cases reaction rates become mass-transport limited. The high viscosity might impede the diffusion of long chains or even of monomers. We come to this point when considering radical polymerization.
Another aspect that is of practical importance, but is often underestimated, is the high demand for purity of the involved chemicals. The concentration of active species is often rather low; in radical polymerization, the concentration of growing chains is mol/l, the concentration of active species in transition-metal-catalyzed polymerization is just one or two orders of magnitude higher. Moreover, in some cases (living polymerization and step-growth polymerization, see Section 2.4.1), the resulting molecular mass is strongly dependent on stoichiometry; here, small errors may prevent us from reaching the desired values.
1.2 Structures in Polymers and Their Influence on Processing and Application Properties
Polymers are used not because they have a certain structure, but because they have certain properties in their final application or during processing. However, the results of kinetic or process models are usually not these final properties but information about the molecular structure of the macromolecules. In the following, we briefly discuss possible structural differences of polymers and some consequences of their properties. This is by far not an exhaustive overview and is greatly simplified. It shall more serve as an appetizer and motivator to use modeling methods to design polymer structures.
1.2.1 Chain Length, Molecular Mass, Moments, and Mean Values
In Section 1.1 we have already pointed out that one important structural property of a macromolecule is its degree of polymerization resp., its molar mass. Synthetic macromolecules usually have a distribution of degrees of polymerization resp. molar masses.
1
Definition 1
By we denote the concentration of chains of degree (chain length) of polymer at time . Therefore, this quantity is related to the number of molecules of a certain kind. (Figure 1.1a,c). The distribution is called frequency or number distribution.
It can also be represented in terms of the molecular mass of the respective chains (Figure 1.1b,d) and as differential (Figure 1.1a,b) or integral distribution (Figure 1.1c,d). Very often, however, one is interested in the mass of polymer molecules. The mass of a chain consisting of single molecules is given by the weighted sum of these units. Let denote the average molecular mass per monomer unit (AMW) in all chains of type . In the simplest case of homopolymers, is just the molar mass of the monomer. We remark, however, that in complex systems, for example, copolymerization, may be a time-dependent function of polymer composition (see (2.287)). Then, the mass of a chain is . Therefore, the weight or mass distribution of mass of a polymer is given by
We have to keep in mind that all these distributions are functions of a discrete variable, as the degree of polymerization can only have integer values . For long chains (), we could deal with them as continuous functions; however, this requires additional assumptions and in view of the fact that this problematic approach is not allowed for short oligomers, we will not discuss it here.
Figure 1.1 Possible representations of distributions: (a) discrete, differential frequency, or number distribution of degree of polymerization, (b) continuous differential frequency or number distribution of molar mass, (c) discrete, cumulative distribution of degree of polymerization, (d) continuous, cumulative distribution of molar mass.
A third important representation of a distribution is induced by the measurement of the molar mass distribution by gel permeation chromatography (GPC). It has been shown [5] that the GPC data are proportional to a distribution
The meaning of this distribution is less intuitive than the mass distribution, but it is apparent that the concentration of long chains is amplified by the factor . In many modeling projects, it is crucial to analyze all three types of distributions. We also note that the numerical approximation of can be much more challenging than to obtain the basic frequency distribution.
In many cases, distributions are just characterized by some average values. For that we introduce the statistical moments of type and of distribution at time
Remark 1 (moment notation)
There are very different notations in use for moments, often the Greek letter instead of is used. The summation bounds are often omitted as well as the superscript, if there is only one distribution being considered.
Remark 2 (moment meaning)
The meaning of the zeroth moment is the total concentration of polymer chains . The meaning of the first moment is the total concentration of all monomer units in all chains of polymer .
Using the first moment of a given polymer distribution and the average molecular mass per monomer unit , the mass concentration of a polymer species can be described by
In some models, particularly in polycondensation, each single polymer chain may carry or (miss) one additional piece of mass, a fragment, of molecular weight . Then the expression (1.6) has to be extended by a term that multiplies the total number of polymer chains with the mass of the fragment. Note that is possible and allowed here.
Next, we use the statistical moments to define some important mean values. The number average describes the average number of monomer units per chain. The mass average leads to the average number of monomer units in a macromolecule to which a randomly chosen monomer unit belongs.
Remark 3
We use the capital letter to characterize the average values instead of the often used notation or , as we have reserved the letter to describe the polymer.
Remark 4
We have to add the superscript in these definitions, since in nearly all situations of interest we have to distinguish different types of polymers. However, outside a model consideration and without a concrete polymer, we will use just .
Often, people are more interested in the corresponding mass averages, only extended by a factor given by the average molecular mass of a monomer unit:
The important width of the distribution is characterized by the polydispersity index (PDI)
Note that PDI can be correlated to the standard deviation usually used in statistics to characterize the width of a distribution by
1.2.2 Rheological Properties
Many of the mechanical and rheological properties of polymers come from the fact that in contrast to small molecules, macromolecules do not exist as separated molecules (except in highly diluted solutions), but are more or less “entangled” (see Figure 1.2). These entanglements act as physical cross-links which drastically hinder the mobility of polymer chains and consequently influence all properties that are connected to chain mobility. Solid-state properties like tensile strength and impact strength usually increase with . In contrast to this, the rate with which a crack propagates in a polymeric material is reduced for higher molecular masses. So, in terms of mechanical properties, high molar masses are desirable. However, the entanglement is also responsible for the extreme high viscosity of polymer melts and concentrated polymer solutions. Viscosity is the ratio of shear stress and shear rate , for example, the resistance of a liquid to replacement. In “normal” liquids, according to Newton's law , viscosity is a constant. For polymer melts and highly concentrated solutions or other more complex liquid systems like dispersions, suspensions, and so on, this simple relation is no longer valid, but becomes a function of . The mostly observed phenomenon with polymer systems is the so-called shear-thinning behavior, where viscosity decreases with shear rate, because by shearing more and more entanglements are released. This is sketched in Figure 1.3.
Figure 1.2 Schematic entanglement of bulk polymers.
Figure 1.3 Shear viscosity curves for polyethylene with different weight average molar mass.
Here, the viscosity at low shear rates, the zero-shear viscosity, correlates with very strongly; above a critical chain length from which on entanglement occurs, the correlation holds for nearly all linear polymers. However, not only the average values influence the shear viscosity but also the distribution itself (Figure 1.4). Broader distributions usually show a stronger shear-thinning behavior, which makes processing easier while maintaining a high molecular mass for mechanical properties.
Figure 1.4 Shear viscosity curves of polyethylene grades with different molar mass distributions (normalized to same zero-shear viscosity).
There are far more properties which depend on chain length, like glass transition temperatures , melting point of semi-crystalline polymers, viscoelastic behavior, and others.
1.2.3 Constitutional Isomers
As in low molecular compounds, in macromolecules also all kinds of constitutional isomers may occur.
If the monomer unit has an asymmetric carbon atom, the macromolecule may show stereoisomerism. It may consist of isotactic sequences, where the substituent is always on one side of the plane, syndiotactic sequences where the position of alternates, or atactic sequences with no regularities. Stereospecific polymers often result from transition-metal-catalyzed polymerization like isotactic polypropylene with Ziegler catalysts. The degree of stereoregularity has a strong effect on melting point, degree of crystallization, or glass transition temperature, and so influences mechanical and optical properties. It is usually characterized by the concentration of the respective triades or pentades.
Geometric isomers may occur, for example, during the polymerization of dienes, when there are several possibilities to open a double bond as in the polymerization of butadiene depending on the process and catalysts/initiators. This results in 1,4-cis, 1,4-trans, or 1,2-polybutadienes with remarkably different properties. Polybutadienes produced with Co- or Ni-catalysts show 95% 1,4-cis content with C and are partly crystalline with C, high 1,2-polybutadiene produced with Li-organyls shows 1,2-content of 90% and has C and C. In emulsion polymerization, one may obtain mixed structures with 10% 1,4-cis, 70% 1,4-trans, and 20% 1,2-structures and C. Consequently, they differ in their application properties as tire rubbers in abrasion resistance, rolling resistance, road holding, and so on.
Other structural isomers may occur with nonsymmetric monomers from the orientation during the addition of the monomer to the active center. There is an agreement that the carbon atom with the larger substituent is called the head of the monomeric unit, and for asymmetric monomers there exist three possibilities of orientation. Head–tail is the “normal” orientation, and irregularities like the head–head or tail–tail addition usually influence properties like the degree of crystallinity.
1.2.4 Architectural Isomers
Until now we have considered macromolecules to be linear molecules; but in many cases, this is not true. There might exist a wide variety of different architectures with the same molar mass.
Comblike or graft polymers:
Starlike polymers:
Treelike polymers:
(long-chain) Branched polymers
Figure 1.5 shows how the architecture may influence properties. The viscosity of starlike polymers is lower than that of the linear polymer with the same molar mass, because the coil radius of a starlike polymer is smaller than that of linear molecules of the same molecular mass.
Figure 1.5 Schematic dependence of melt viscosity on molar mass for different architectures.
A special class of polymer architectures are cross-linked polymers. Polymer chains in a network have at least two cross-linking points by which they are connected to other chains of the network; so, a polymer network, in principle, consists of just one molecule. Polymer networks may be built during the polymerization process itself or may be formed starting from linear chains which then will be cross-linked in a separate process (vulcanization). It is a special challenge to describe their structural properties, like the concentration of cross-links, the chain length between cross-links, and so on. Depending on the degree of cross-linking, polymer networks may be soft, elastic, and swellable (like superabsorbent polymers, rubber tires), or they may be hard and brittle (as in the case of phenol-formaldehyde resins, Bakelite®, the earliest commercial synthetic resin)
1.2.5 Copolymers
Only a few commercial polymers consist of just one monomer; most of them are copolymers of two or more monomers, . The obvious difference between copolymers is their composition, that is, the fractions of the monomer in the polymer. However, copolymers of the same overall composition may differ in the sequence of the different monomers along the chain. Most common are so-called “statistical” copolymers where the monomer units are randomly distributed along the chains. We see in Section 2.6 that they obey certain statistics; for example, they can be treated as Markov chains of zeroth, first, or second order. Strongly alternating copolymers can be obtained if neither of the two monomers is able to form a homopolymer. Finally, block copolymers may be formed by sequential addition of different monomers to living initiators (see Chapter 2) or by coupling of separately formed homopolymers.
Statistical copolymers:
Alternating copolymers:
Block copolymers:
It is obvious that these different structures may cause tremendous differences in the physical and application properties of copolymers. This is exemplified in Figure 1.6 for the glass transition temperature of a block and a statistical copolymer. Block copolymers show two glass transition temperatures, being those of the two homopolymers. Statistical copolymers only show one glass transition temperature that is somehow the average of the glass transition temperatures of the homopolymers.
Figure 1.6 Typical glass transition temperature behavior of statistical and block copolymers.
There are several characteristics by which the monomer sequence along the chain can be described. As with the degree of polymerization, we also have to deal with distributions, so there are characteristics describing average values like the average sequence length of monomer describing how many units (on average) follow one after the other without being interrupted by another monomer. There are several methods to take into account the distribution properties. One possibility is to use the sequence length distribution, which gives the fraction of sequences with units . Another possibility to look at the distribution is to characterize a copolymer according to the fraction of triads or pentads around the central unit (see Figure 1.7), that is, what are the direct neighbors of or even the next but one neighbor.
Figure 1.7-centered triads and pentads.
We come to these characteristics in more detail in Section 2.6. Furthermore, naturally also for copolymers, there may exist architectural isomers, as described in Section 1.2.4. Here, the main and side chains may all consist of statistical copolymers, or the main and side chains may consist of different polymers, or any other possibility as shown here
Therefore, the introduction of more than one monomer increases the number of possibilities for the structure of macromolecules dramatically. A simple example will demonstrate this. Imagine a copolymer consisting of equal moles of two monomers, for example, styrene (S) and methyl methacrylate (MMA), and the distribution of degrees of polymerization of this polymer has a number average of . Now let us ask how many macromolecules with this average chain length 1000 and the average composition of 500 styrene and 500 MMA units may exist. This is equivalent to the question of how many possibilities do exist to put coins into places (irrespective of the sequence). From combinatorial analysis, we know that this number is
For our example and in view of symmetry, this yields possibilities. The molar mass of such a chain (, /mol) then is 102,000 g/mol, and we get the absolute mass of one chain as g by dividing the molar mass by . Thus, if we had the task of synthesizing one molecule for all of the possibilities, we end up with the production of t. The mass of our galaxy is assumed to be in the range of t. So, it is rather unlikely that we ever have produced two identical molecules of this kind. Moreover, here we just have considered one kind of isomerism, that is, the positioning of the monomer. The other types of isomerism mentioned will amplify the possibilities of how a macromolecule with a definite chain length (=molar mass) and a certain composition may look like in all detail.
This shows that especially for synthetic polymers, it is difficult to look at individual molecules. We should better characterize them by some averaged quantities; some of them have been introduced in this chapter, and they – and additional ones – are described in detail when appropriate. Nevertheless, there might be cases where we will have a look at individual species (see Section 5.5).
1.3 Some Analytical Methods for Model Validation
The main purpose of the models we are dealing with is to link information about the structure of the macromolecules and the process for producing these molecules. The mathematical tools are important, but at least as important for a successful modeling project is to have analytical tools at hand which give us the structural information about the polymer. This experimental information is important for two reasons. In the beginning of a project, it will help us find a proper estimate of the model parameters – rate coefficients, distribution coefficients, and so on. In the validation phase, experimental data are used to proof the quality of the model. Table 1.2 gives some hints for possible analytical methods, but it is not exhaustive; and, in many cases, the suitable method depends on the system under investigation.
Table 1.2 Methods for determining macromolecular structures
Model output
Measurement method
Class
Remarks
Vapor pressure osmometry
A
20 kg/mol
Membrane osmometry
A
100–1000 kg/mol
Cryoscopy, ebullioscopy
A
10 kg/mol
End group analysis (spectroscopy, titration)
E
30 kg/mol
Static light scattering
A
10 kg/mol, gives radius of gyration
Dynamic light scattering
A
10 kg/mol, gives hydrodynamic radius
Small-angle X-ray scattering, SAXS
A
10 kg/mol
Small-angle neutron scattering, SANS
A
10 kg/mol
Solution viscosimetry
R
10 kg/mol
Molar mass distribution
Gel permeation chromatography, size exclusion chromatography
A (R)
Depending on detector
Molar mass distribution
Analytical ultracentrifuge, sedimentation velocity, sedimentation equilibrium
A
Diffusion and sedimentation coefficient
Molar mass distribution
Field flow fractionation
A (R)
Depending on detector
Molar mass distribution
MALDI-TOF
A
Molar mass distribution
Dynamic mechanical analysis (DMA)
R
Chemical composition
Spectroscopic methods, elementary analysis
A
Depending on system
Sequence lengths, triades, pentades in copolymers
Spectroscopic methods
A
Depending on system
Bivariate distribution in copolymers
2D chromatography (HPLC+SEC)
A, R
Tacticity
-NMR, FT-IR
A
Degree of short chain branching
NMR, FT-IR
A
Degree of long-chain branching
Solid-state-NMR, FT-rheology, SEC with triple detector (RI, LS, viscosimetry)
A
Cross-linking density
Swelling
R
Model gives chemical, analytics often chemical + physical cross-linking
Cross-linking density
Elasticity modulus, shear modulus, DMA
R
Model gives chemical, analytics often chemical + physical cross-linking
When looking at the various methods, we can classify the measurement methods into three classes (see Table 1.2):
1.
Absolute methods (A) give the property without any assumptions about the chemical or physical structure of the molecule. Typical absolute methods are, for example, those measuring colligative properties (vapor pressure osmometry, membrane osmometry).
2.
Equivalent (E) methods need some assumptions about the chemical structure of the molecule like for example end group titration.
3.
Relative methods (R), in all cases, need calibration as they depend on the chemical structure of the solute and its interaction with the solvent, like viscosimetry.
