computer simulations of poly(ethylene oxide): force field, pvt diagram and cyclization behaviour

Polymer International 44 (1997) 311È330

Computer Simulations of Poly(ethyleneoxide): Force Field, PVT Diagram and

Cyclization Behaviour¤

David Rigby,* Huai Sun & B. E. Eichinger

Molecular Simulations, Inc., 9685 Scranton Road, San Diego, CA 92121, USA

(Received 17 March 1997 ; accepted 17 June 1997)

Abstract : Parametrization of a force Ðeld capable of quantitatively describingthe gas, liquid and crystal phases of alcohols, ethers and polyethers is described.Two applications are reported, the Ðrst employing atomistic simulations to studyPVT (pressure, volume, temperature) and cohesive properties of oligomericpoly(ethylene oxide) (PEO) and related small-molecule liquids, and the second tostudy the extent of ring formation in polymerization of poly(ethylene glycols)(PEGs) and hexamethylene diisocyanate (HDI). The atomistic simulations, focus-ing extensively on liquids and amorphous poly(ethylene oxides), demonstrate theability to predict densities with an accuracy of 1%È2% over extended ranges ofat least 200 K in temperature and 180 MPa in pressure. Densities of relatedsmall-molecule liquids, dimethyl and diethyl ether and ethanol at or close tosaturation pressure are also well reproduced to temperatures close to the criticaltemperature. Densities calculated for methoxy-terminated oligomers are used topredict the density of melt and amorphous high-molar-mass PEO with an accu-racy of better than 1%. Similarly, solubility parameters have been calculated as afunction of chain length for poly(ethylene glycol) oligomers and used e†ectivelyto obtain estimates of the solubility parameter of high-molar-mass material.Additionally, crystal structures can also be well predicted. For the poly-merization studies the Monte Carlo network simulation method was modiÐed tomimic di†usion of reactants during the polymerization. Application to thePEG/HDI “linearÏ polymerization system, using chain conÐgurations generatedwith the atomistic force Ðeld, reveals a major improvement in the ability of themethod to predict the extent of ring formation without adjustable parameters forpolymerization conditions ranging from the bulk to highly dilute reaction condi-tions.

Polym. Int. 44, 311È330 (1997)No. of Figures : 11. No. of Tables : 8. No. of References : 69

Key words : polyethers, alcohols, ethers, PVT diagram, cohesion, cyclization.

* To whom all correspondence should be addressed.¤ Dedicated to Professor Bob Stepto on the occasion of his 60th birthday.Contract grant sponsor : Office of Naval Research ; Contract grant number : N00014-95-C-0156.Contract grant sponsor : ARPA; Contract grant number : F30602-95-2-0007/C-5-2779/AO C277.

3111997 SCI. Polymer International 0959-8103/97/$17.50 Printed in Great Britain(

312 D. Rigby, H. Sun, B. E. Eichinger

INTRODUCTION

The industrial applications of ethylene-oxide-containingpolymers are extensive. They include cosmetics, phar-maceutical formulations, dental adhesives, lubricants,detergents, Ñocculants, urethane elastomers, solid elec-trolytes and Ðlms.1 While in some ways the materialsused in these applications vary widely in properties (e.g.with molar masses typically O(102)ÈO(107)), they alsoshare an important common characteristic, namely thatthe polymer is used either in the liquid or predomi-nantly amorphous state or in solution. As a conse-quence, some of the material properties of criticalimportance are cohesive properties and PVT behaviour,and conformational properties of individual chain mol-ecules. Thus, for example, cohesive properties determinecompatibility in PEO-containing blends and solutions,PVT behaviour is of particular interest in modelling ofpolymer processing, and conformational behaviourdetermines various properties such as solution vis-cosities and the extent of cyclization/loop formation inlinear and non-linear step growth polymerization reac-tions.

Early modelling studies of poly(ethylene oxide) (PEO)and poly(ethylene glycols) (PEGs)Èthe common termfor hydroxyl-terminated oligomersÈfocused on predict-ing the inÑuence of molecular structure and chain con-formational preferences on experimentally measuredproperties such as dipole moments in solution,2 trans-lational di†usion,3,4 temperature dependence of chaindimensions5,6 and packing in the crystalline state.7 Themajority of such studies utilized the rotational isomericstate approach,8,9 parametrization of which was thesubject of extensive studies in the 1960s and 1970s.5,10In recent years, rapid advances in speed and availabilityof computers have led to renewed interest in modellingof PEO, both with a view to improving the quality ofthe RIS treatment and, increasingly, with a view to per-forming direct simulations of the bulk phase. To cite afew examples : Smith et al.11,12 have applied high-levelab initio calculations to develop a third-order RIS treat-ment and to parametrize a force Ðeld for simulating 1,2-dimethoxyethane and PEO; Mu� ller-Plathe and vanGunsteren13 have studied the e†ects of polar environ-ments on conformer populations ; Engkvist andKarlstro� m14,15 have used a Monte Carlo approach tostudy temperature and concentration dependencies ofconformer populations of dimethoxyethane and PEOoligomers ; and Neyertz and co-workers16,17 havereported Monte Carlo and molecular dynamics studiesof PEO conÐguration and structure in both the crys-talline and melt states.

Within our own laboratory, major complementaryareas of study over the past few years have involvedsimulations at both the atomistic and coarse-grainedlevels. The atomistic simulation research is aimed atachieving two major objectives. Firstly, it is desired to

develop a high-quality force Ðeld capable of quantitat-ively describing the condensed phase properties of poly-meric and organic materials to the optimum extentpossible using current force Ðeld technology. Secondly,it is desired to generate a comprehensive set of baselineproperty data suitable for evaluating future develop-ments in modern, largely ab initio, force Ðeld technol-ogies. The coarse-grained work, involving Monte Carlosimulations, concentrates on problems at di†erentlength scales, in which it is aimed to predict the devel-opment of topology which occurs during linear andnon-linear polymerization reactions.

The remainder of the present paper is divided intothree major sections as follows. Section 2 reviews prin-cipal issues in atomistic force Ðeld development forethers and alcohols structurally related to poly(ethyleneoxides) and gives a detailed description of force Ðelddevelopment and initial validation results. This is fol-lowed in Section 3 by an extended study in which theforce Ðeld is applied to calculate crystal structures andthermophysical and cohesive properties of liquids.These studies are of two types, namely utilizing mol-ecules used in the parametrization but subject to condi-tions not considered in the parametrization, as well as asecond class of mostly oligomeric molecules notincluded in the parametrization. Finally, Section 4describes a further application in which conformationalproperties of oligoethylene glycols are computed usingthe force Ðeld and used in combination with a recentmodiÐcation of the Monte Carlo network simulationmethod of Eichinger and co-workers18,19 to predict theextent of ring formation in linear step growth poly-merization reactions involving PEG oligomers andhexamethylene diisocyanate. Results are then comparedwith earlier experimental measurements reported andanalysed by Stepto and co-workers.20,21

2 FORCE FIELD PARAMETRIZATION

2.1 Overview of force field development of alcoholsand ethers

As two of the most common classes of organic com-pounds, alcohols and ethers have been parametrized inseveral force Ðelds such as MM3,22 AMBER23 andCFF93 (see Ref. 24 for a review of the CFF93 para-metrization methodology). For small linear or cyclicalcohol or ether molecules these force Ðelds yield rea-sonably good agreement with experimental molecularstructure data, conformational energies and vibrationalfrequencies. Although in principle these force Ðelds canbe used for simulation of polymers containing the samefunctional groups, a few additional issues need to beaddressed.

One of the major issues concerns the conformationalproperties of poly(ethylene oxide). Extensive experimen-tal and theoretical studies on conformational behaviour

POLYMER INTERNATIONAL VOL. 44, NO. 3, 1997

Computer simulations of poly(ethylene oxide) 313

of related molecules such as 1,2-dimethyoxyethane havebeen reported in the literature. For a long time the con-formational preferences of this molecule were a contro-versial issue. However, recently published resultsemploying high-level ab initio calculations25,26 withlarge basis sets indicate that the TGG@ conformer isonly a fraction of a kcal mol~1 higher in energy thanthe most stable conformer TTT. Although the actualvalues of the calculated conformational energies, whichare highly sensitive to the basis set and underlying cal-culation method, are somewhat arguable, it is agreedthat the intramolecular non-bond interactions betweenthe methyl group and ether oxygen are clearly the factorresponsible for the complicated conformational behav-iour found in this molecule and in related polymers.

As stated earlier, since most applications involvepolymers in the condensed state, in order to be able tomake accurate predictions of the properties of thesematerials using atomistic simulations, there is a need forthe development of force Ðeld parameters capable ofquantitatively describing the condensed phase behav-iour. Most well-parametrized force Ðelds such as thosedescribed in Refs 22È24 are able to predict molecularstructures, conformational properties and vibrationalfrequencies for isolated molecules, but very few have sofar been optimized to model molecules in condensedphases. With the fast advance of computer hardware, abinitio calculation methods and force Ðeld developmenttechniques it is possible today to develop a more accu-rate and Ñexible force Ðeld that can be used to predictproperties for molecules in di†erent environments,including the solid, liquid and gas phases. The recentlypublished OPLS all-atom force Ðeld27 is one example ofsuch developments.

In our research group we have been using ab initiocalculations and experimental condensed phase pro-perty data to parametrize an accurate all-atom forceÐeld for computer modelling of common organic andinorganic molecules and polymers. Some of the resultshave been presented in recent publications onpolysilanes28 and polysiloxanes.29 In the remainder ofthe present section we present another important partof this work, namely the parametrization and validationof a consistent force Ðeld for molecules and polymerscontaining alcohol and ether functional groups.

2.2 Parametrization methodology

2.2.1 Introduction. The functional forms used in thepresent force Ðeld are those used by the consistent forceÐeld (CFF).24 The energy expression, given explicitly inRefs 24 and 29, contains functions which can be broadlydivided into two categories, namely valence and non-bond interactions. The valence terms include terms fordescribing (non-harmonic) bond stretching, anglebending, torsion and out-of-plane motions respectively,and cross-coupling terms between internal co-ordinates,

which have been shown to be necessary for predictingvibrational frequencies and structural variations associ-ated with conformational changes.30 The non-bondinteractions, which include Lennard-Jones (LJ) andelectrostatic terms, are used for interactions betweenpairs of atoms separated by two or more interveningatoms, or those that belong to di†erent molecules. A9È6 function is used to represent the LJ interaction.31The electrostatic interaction is represented by thepartial atomic charge model, using bond increments

deÐned so that the net charge for atom i is a sum-dij

30mation of all bond increments related to this atom, i.e.

qi\ ;

jdij

In the original CFF-type force Ðeld24,30,32 the non-bond parameters were determined mainly based on atechnique utilizing crystal data, as developed in the1960s and 1970s by several research groups (see Ref. 33for a general review). Normally, this is done via energyminimizations, with the internal co-ordinates Ðxed attheir experimental equilibrium values and the non-bondparameters treated as adjustable parameters to Ðt thecrystal structure, lattice energies and some latticedynamics properties.34

Since the experimental observables are measured atÐnite temperatures, whereas the parameters are derivedbased on energy minimizations (equivalent to a tem-perature of 0 K), parameters derived using thisapproach implicitly contain thermal expansion andquantum e†ects at the given temperature. Consequently,good agreement between subsequent calculations andexperiments can only be expected when (i) the calcu-lations are performed using energy minimizations, and(ii) the experimental data are measured under condi-tions that closely approximate those used in the para-metrization. Generally speaking, such parameters arenot capable of predicting properties over a range ofconditions (e.g. of pressure and temperature).

In aiming to develop a Ñexible force Ðeld which yieldsgood agreement between calculations and experimentalmeasurements under a broad range of application con-ditions, we have adopted a revised approach whichcombines the CFF method of deriving intramolecularvalence parameters and electrostatic non-bond inter-actions with an empirical method35,36 based on molec-ular dynamics (MD) simulations of liquids toparametrize the van der Waals non-bond parameters.Since pressure, density and temperature are explicitlyincluded in the MD simulations, parameters derivedusing this approach are independent of the thermophy-sical conditions. Hence they are closer to physicalreality than those derived using the minimizationscheme.

2.2.2 Parametrization procedures. To represent the dif-ferent atoms in alcohol and ether functional groups,



seven atom types are used. These atom types includethree for carbons, two for hydrogens and two foroxygens, as given in the Appendix. Two carbons,denoted c4 and c3a, and a generic non-polar hydrogenatom type, h1, were transferred from alkyl and phenylgroups parametrized earlier.37 In the present work, pre-liminary studies indicated the necessity to introduce anew atom type for carbons alpha to ether or alcoholoxygen in order to obtain an overall good Ðt of liquidproperties for molecules with di†erent sizes. For asimilar reason, two atom types are used for oxygens,namely o2h for hydroxy oxygen and o2e for etheroxygen. Physically, this is a consequence of the shift inelectron density towards the oxygen in a CwO bondpair. Finally, the polar hydrogen (hydrogen bonded tooxygen) is represented by h1o.

These atom types are “formalÏ atom types used in theforce Ðeld. For each of the interaction terms (non-bond,bond, angle, torsion and out-of-plane bending), more“genericÏ atom types may be used so that the totalnumber of parameters can be greatly reduced withoutlosing accuracy. Particularly, for valence terms, it isoften found to be reasonable to utilize some genericatom types. This is implemented using an “equivalencetableÏ linking the formal atom type with actual atomtypes used in the force Ðeld. This table is included in theAppendix. For example, c4o is equivalent to c4o for thenon-bond term and to “genericÏ c4 (an alkyl carbon) forall other terms ; similarly, o2e is linked to itself for non-bond and bond terms, including both bond stretchingand bond increments, but to o2 for angle, torsion andout-of-plane terms. For non-bond terms the formalatom types are equivalent to the actual ones.

The valence parameters for alcohols and ethers weredeveloped earlier as part of the CFF93 force Ðeld.24Similar to alkanes, carbonates, etc., whose parametriza-tion work has been published elsewhere,32 parametersfor alcohols and ethers were derived based on ab initioHF/6È31G* data and subsequently scaled to Ðt theexperimental data for molecular structures, conforma-tional energies and vibrational frequencies. It was foundthat these parameters were able to predict these molecu-lar properties with reasonable accuracy. Hence mostvalence parameters were initially transferred fromCFF93. However, it was necessary to make some modi-Ðcations, particularly of the torsional terms, for tworeasons. Firstly, since modiÐcations were to be made tothe non-bond parameters (i.e. atomic partial chargesand Lennard-Jones terms) and since the non-bond andsome of the valence terms are strongly coupled, modiÐ-cation of valence terms has to be made to compensatefor any change in the non-bond terms. Secondly, theoriginal parameters were derived based on ab initio cal-culations at the HF/6È31G* level. It has subsequentlybecome clear that there is a signiÐcant discrepancy inthe calculated conformational properties between theseand high-level calculations. Accordingly, in the present

study the latest ab initio results25,26 were used to Ðx thetorsional parameters.

Using MD simulations of liquids to parametrize non-bond parameters is a very time-consuming process. Tomake this project manageable, the number of adjustableparameters has to be controlled and kept to a minimumreasonable level. This was achieved by (i) parametrizingmost internal parameters using ab initio data and (ii)transferring “knownÏ parameters, which in this caserefers to all parameters previously derived for alkyl andphenyl groups. In addition, prior to parametrization ofthe van der Waals parameters using MD simulations,the charge parameters and valence parameters werecarefully examined and Ðxed. Hence the adjustableparameters for the MD simulations were reduced to theLennard-Jones parameters associated with the c4o, o2h,o2e and h1o atom types.

The CFF93 charge parameters for alcohols andethers were originally derived by Ðtting molecularcrystal structures and dipole moments. In recent yearswe have instead made use of the ab initio electrostaticpotential (ESP), which represents the intermolecularelectrostatic interactions more accurately, to para-metrize the charge parameters.38 Consequently, tomake the parameters consistent with those derived forother functional groups studied in our recent work, theCFF93 charge parameters for alcohols and ethers werereplaced with a newly derived set obtained using theESP approach. To ensure transferability, the ab initioESP surface was Ðtted subject to a constraint that Ðxesthe “knownÏ parameters such as bond charge incrementsof c4Èh1 and c3aÈh1 taken from previous work. Theresulting bond increments are given in the Appendix.

After determining the charges, the valence parameterswere examined carefully by comparing calculatedmolecular structures, conformational properties andvibrational frequencies with experimental data. Thisstep is crucial because of the coupling between theintramolecular and intermolecular interactions. Forexample, deviations in bond lengths and angles have adirect impact on the molecular volume and, conse-quently, the volume of the bulk. For any error that waslarger than a nominal value (1% for the bond lengthsand angles), the most sensitive parameters were identi-Ðed and modiÐed to reduce the error.

With the intramolecular (valence) properties andintermolecular electrostatic interactions well represent-ed, the Lennard-Jones parameters were then subjectedto stepwise modiÐcation using MD simulations ofliquids. For each set of LJ parameters, NVT moleculardynamics simulations were performed and the pressureand cohesive energy were calculated. This process wasthen repeated to Ðnd the set of values which best repro-duces the experimental cohesive energy density andmass density at atmospheric pressure and ambient tem-perature.

The molecular dynamics simulations used in the



reÐnement procedure were performed using cubicperiodic cells containing typically 1000È2500 atoms, asize which is generally sufficient for system size e†ectson pressure and cohesive energy to be negligible.29 AllLJ parametrization calculations were performed usingNVT ensembles, in which temperature control wase†ected using the stochastic collision approach devel-oped by Andersen and co-workers,39a with collisionsapproximately every 300 fs. Integration of the equationsof motion was performed using the velocity form of theVerlet algorithm,39b with an integration times step of1É0 fs. Moreover, initial checks were made to ensurethat, for simulation under the ambient conditions usedfor parametrization (frequently c. 273È303 K and zeropressure), simulations with shorter, 0É5 fs, time stepsgenerated indistinguishable trajectories and physicalproperties. Following the parametrization preliminaryvalidation work was performed using simulations in theNPT ensemble, with temperature again controlled usingthe stochastic collision approach and with pressure con-trolled using the loose coupling method of Berendsen etal.,40 with the characteristic decay time qB 150 fs. Con-sistency between calculations in the two ensembles wasveriÐed by Ðrst performing NPT simulations at a speci-Ðed pressure to obtain the density, followed by an NVTsimulation at the predetermined density to conÐrm thatthe average pressure agreed with that used in the NPTrun.

Both LJ and electrostatic interactions are treatedusing a group-based approach, in which the moleculesare Ðrst partitioned into small, electrically neutralgroups (such as CH, etc., asCH2 , CH2OCH2 ,required), with one of the atoms close to the geometriccentre designated as the switching atom. Each non-bond calculation for atom i then includes interactionswith all atoms in groups which are neighbours of thegroup to which atom i belongs. In turn, neighbouringgroups are deÐned as those groups whose switchingatoms are separated by distances less than a cut-o†value, usually in the range This technique8È14 Ó.avoids artefacts associated with so-called “splittingÏ ofdipoles, which would otherwise occur using the simplerpractice of evaluating non-bond interactions for allatoms separated by less than a speciÐed cut-o† distance.In addition, selected calculations have occasionally beenrepeated, evaluating the electrostatic interactions usingthe more expensive Ewald summation approach tocheck for consistency.

Finally, since applying cut-o†s to the van der Waalsinteractions introduces an error in the calculated valuesof both energy and pressure,29,41 the computed valueswere corrected using standard methods. This correctionand all other calculations described herein were per-formed using the DISCOVER simulation package.42

After the modiÐcation of the non-bond LJ param-eters, in principle, some valence parameters may need tobe adjusted to compensate for the changes, and the

whole process may need to be repeated. In practice,however, with initial non-bond parameters of reason-ably high quality, the impact of the new LJ parameterson the valence parameters was generally found to besmall. In many cases it was found that only torsionterms are a†ected. Consequently, one iteration wasoften found to be sufficient, because the couplingbetween the LJ and torsion parameters is unidirec-tional. In other words, when the LJ parameters arechanged, the torsion parameters may require someadjustment, but a modest change in torsional param-eters normally does not a†ect the calculated ther-mophysical properties. Consequently, there is no needto reÐne the non-bond parameters again.

2.3 Initial force field validation

2.3.1 Intramolecular properties. The Ðnal parametersobtained after applying the above methodology to alco-hols and ethers are given in the Appendix. Using theseparameters, molecular mechanics calculations were per-formed for a group of small alcohol and ether mol-ecules. The calculated molecular structures,conformational energies and vibrational frequencies aresummarized in Tables 1È3. It can be seen that themolecular structures are well represented and that theconformational energies are generally in good agree-ment with the experimental data or high-level ab initioresults. The calculated vibrational frequencies are inreasonable agreement with the experimental data,although some of the deviations are larger than

TABLE 1. Comparison of molecular structures

Exp. Exp. Calc.

Ethanol Ref. 58 Ref. 59

CwC 1É511 5 1É529 7 1É522CwO 1É431 0 1É424 7 1É423CwH 1É088È1É098 1É093 6 1É104OwH 0É971 0 0É945 1 0É950CCO 107É8 107É3 107É7COH 105É4 108É5 107É4CCH 110É1È110É7 110É3 111É6

Dimethyl ether Ref. 60CwO 1É415 1É411CwH 1É118 1É104COC 111É8 110É3HCH 109É2 108É3

Ethyl methyl ether Ref. 61 Ref. 62CwC 1É520 1É521 1É523CmewO 1É413 1É415 1É414CetwO 1É422 1É407 1É419CwH 1É118 È 1É107COC 111É9 111É4 111É9OCC 109É4 108É9 107É1HCH 109É0 È 108É4



TABLE 2. Comparison of experimental and calcu-

lated conformational energies

Exp. Calc.

Ethyl methyl ether Ref. 24a

DE(0) 0·0 0·0

(120) 2·67 2·29

(0) 7·55 5·95

Gauche 1·92 (70·0) 1·44 (72·3)

Ethanol Ref. 63

DE tÉ¿g 0·4 0·31

CwC barrier 3·08 3·16

Dimethyl ether Ref. 64

DE(barrier) 2·72 2·67

Dimethoxyethane Ref. 26

tgg 1·51 1·46

tg½gÉ 0·23 0·26

tgt 0·14 0·13

ttg 1·43 1·42

ttt 0·00 0·00

a ÍFF9Ë parameters for ethers and alcohols : unpublished

work.

obtained using the original CFF parametrizationmethod.

2.3.2 L iquids. NVT and NPT simulations were per-formed with Ðnite group-based cut-o†s and with(8É5 Ó)tail corrections for van der Waals interactions. The timestep used was again 1 fs. All systems were equilibratedunder the simulation conditions for at least 60 ps beforecollecting the trajectories for subsequent analysis. ForNVT simulations, pressures and cohesive energies wereevaluated over the last 50 ps of the simulations ; forNPT simulations, densities were averaged over the last100 ps. The NVT-calculated pressures are given Table 4.For comparison, calculations were performed also usingthe older CFF93 force Ðeld parameters. The data inTable 4 show that with the older parameters the calcu-lated pressures are systematically rather too large.Using the new parameters, the calculated pressures arescattered around the experimental value (D0É1 MPa)with a range of c. ^10É0 MPa. In this context it shouldalso be noted that for a 100 ps NVT simulation on atypical liquid system of the size studied here, normal

TABLE 3. Comparison of experimental and calcu-

lated vibrational frequencies of ethanol

Exp.65 Calc. Diff.

3 676 3 652 É24

2 989 2 979 É10

2 989 2 969 É20

2 989 2 941 É48

— 2 904 —

2 900 2 889 É11

— 1 487 —

— 1 468 —

1 452 1 451 É1

1 452 1 422 É30

— 1 386 —

1 393 1 263 É130

1 241 1 240 É1

1 062 1 125 63

1 033 1 103 70

— 1 020 —

885 850 É35

801 774 É27

419 360 É59

243 316 73

201 223 22

TABLE 4. Pressures calculated from NVT calcu-

lations on various liquid ethers and alcohols

Temp. (K) r Pold

Pnew

Ethanol 293·2 0·789 504 É22

Propanol 293·2 0·804 767 113

Dimethyl ether 248·2 0·735 642 81

Diethyl ether 293·2 0·714 317 55

pressure Ñuctuations lead to an estimated uncertainty inpressure (standard error in the mean) of c. 3É0È5É0 MPa.Consequently, it would be difficult to concludeunequivocally that the observed pressure deviations arestatistically signiÐcant. Moreover, since typical liquidcompressibilities are of the order of 10~3MPa~1, theobserved pressure deviations would lead to only minordensity uncertainties for simulations performed at con-stant pressure.

The calculated cohesive energies (in kcal mol~1) are

TABLE 5. Cohesive energies (kcal mol—1) calculated for various liquid ethers

and alcohols

Temp. (K) CEDexp

CEDold

% Error CEDnew

% Error

Ethanol 293·2 9·62 9·37 É2·6 9·54 É0·8

Propanol 293·2 10·86 10·31 É5·1 10·87 0·0

Dimethyl ether 248·2 4·65 5·18 11·4 4·57 É1·7

Diethyl ether 293·2 5·97 6·80 13·9 6·11 2·3



TABLE 6. NPT densities (g cm—3) of various liquid

ethers and alcohols

Temp. (K) rexp

rcalc

% Error

Ethanol 293·2 0·783 0·789 0·8

Propanol 293·2 0·794 0·804 1·2

Dimethyl ether 248·2 0·735 0·714 É2·9

Diethyl ether 293·2 0·714 0·706 É1·1

given in Table 5. For each of the calculated values thepercentage error with respect to experimental data isgiven in the table. Using the older parameters, agree-ment with experimental data is poor ; systematically, thevalues are too low for alcohols but too high for ethers.With the new parameters, overall agreement withexperimental data is considerably improved.

Using NPT simulations, average densities were calcu-lated for these alcohol and ether liquids. As shown inTable 6, the computed densities generally agree with theexperimental values.

3 EXTENDED VALIDATION

Extended validation essentially focused on the e†ec-tiveness of the force Ðeld when applied under conditionsnot considered in the parametrization. This includes dif-ferent thermodynamic conditions and physical states, inaddition to di†erent molecules. This section reports onthree types of study as follows.

In the Ðrst study, calculations were performed onsmall-molecule crystals. Two of these, erythritol and tri-oxane, were not used in the non-bond parametrization,while another two, ethanol and diethyl ether, were used.However, in the latter two cases the temperaturesappropriate for crystal studies, 87 and 128 K respec-tively, di†er considerably from the ambient tem-peratures used in the parametrization.

The second group of extended validations focused onPVT properties of low-molar-mass liquids and oligo-meric PEOs over a broad range of temperatures andpressures. As was the case for crystals, some of the mol-ecules studiedÈdimethyl and diethyl ether andethanolÈwere used for the non-bond parametrization,though only under ambient conditions. These threemolecules were studied over temperature ranges of up

to 300 K, including the particularly difficult region ofthe PVT diagram in the vicinity of the critical point.Other molecules studied included selected PEO oligo-mers having either two OH, two or one OH andOCH3one end groups. These had not been consideredOCH3during the parametrization. Also in these cases, simula-tions were performed over a 170 K temperature and180 MPa pressure range. Finally, to examine the limit-ing case of high-molar-mass PEO, the dimethoxy-terminated systems have been studied for a range ofmolar masses up to c. 11 000 g mol~1 (250 repeat units).All simulated chain molecule systems had an M1 w/M1 nratio of 1É00.

The third and Ðnal group of validations turned atten-tion towards cohesive properties, as expressed throughthe solubility parameter. These studies includeddimethyl ether and ethanol as a function of tem-perature, and PEG oligomers (i.e. dihydroxy-terminatedchains) as a function of chain length at room tem-perature.

3.1 Crystals

The results obtained on molecular crystal systems aresummarized in Table 7. Two types of calculations wereperformed, namely energy minimization and moleculardynamics at the experimental measurement tem-perature. The energy minimizations were carried out forthe primary cells, and all degrees of freedom, includingintra and intermolecular co-ordinates and all cellparameters, were relaxed. Also in this case, Ewald sum-mation was used for evaluation of both LJ and electro-static non-bond energies and forces. The MDsimulations were carried out in the NPT ensemble.40To reduce the magnitude of the simulation Ñuctuations,supercells containing sufficient primary cells such thatthe supercell dimensions are larger than were used20 Óin the simulations. Owing to slow convergence of theconstant-pressure simulations, MD runs using theEwald summation are very expensive. Consequently, asan alternative, a Ðnite cut-o† with the tail correctionscheme was used. Although this scheme is not strictlyvalid for crystals in the sense that the atomÈatom paircorrelation functions g(r) do not converge to a value ofunity at large r, with these and other systems studied, itwas found that reliable results can be obtained by prop-erly selecting the cut-o† value so that the assumption of

TABLE 7. Densities of ether and alcohol crystals

Temp. (K) rexp

rstatic

% Error rdynamic

% Error Ref.

Ethanol 87·0 1·025 1·079 5·3 1·036 1·1 66

Erythritol 22·6 1·488 1·533 3·0 1·512 1·6 67

Diethyl ether 128·0 0·952 1·032 8·4 0·972 2·1 68

Trioxane 103·0 1·456 1·525 4·7 1·418 É2·6 69



all g(r)\ 1 is approximately valid owing to cancellationof the alternation found in these functions.

Since the parameters were derived based on MDsimulations, the densities obtained from minimizationsare from 3% to 8% larger than the experimental values.However, with MD simulations under experimentalconditions (1 atm and given temperature) the calculateddensities are in good agreement with experimentalvalues. Consequently, it appears that the force Ðeld ise†ective at predicting crystal structures, with a precisioncomparable with that found for liquids.

3.2 PVT Behaviour of liquids and oligomers

Liquid and oligomer PVT behaviour was investigatedusing NPT dynamics as outlined previously. A typicalrun was of duration 100 ps and was preceded by a short(8È12 ps) NVT equilibration stage. As shown pre-viously,29 this approach gives a standard error in themean of density estimates of the order of ^ 0É3%. Inview of the more extreme conditions of temperature andpressure probed by the extended validation studies, anumber of preliminary checks were made for possiblesources of artefacts originating from the parametersused to control the MD algorithms and energy/gradientcalculations as follows.

First, the e†ect of varying the non-bond cut-o† radiuswas examined for triethylene glycol at 373 K, using thegroup-based cut-o† approach for both electrostatic andLJ interactions and using cut-o† values in the range

As a reference, the density was also calcu-7É5È11É5 Ó.lated using the Ewald summation approach for all elec-trostatic and LJ non-bond interactions. Densitiescalculated using all cut-o† values greater than or equalto gave computed densities within 0É7% of experi-8É5 Óment. Additionally, all cut-o† values of and10É5 Óabove gave values within 0É6% of the Ewald summationresult. Consequently, except for a single case to bedescribed below, it was decided to use the (conservative)

cut-o† throughout the rest of the work.10É5 ÓIn a second series of studies the e†ects of Ðnite time-

step-related integration error on computed pressure(and hence density) were again investigated. Thesee†ects are illustrated in Fig. 1, in which the computeddensity at zero pressure is plotted for dimethyl anddiethyl ether at temperatures ranging from 303É2 K towithin 10È20 K of the critical temperature (400É1 and467É0 K respectively for dimethyl and diethyl ether).Here the full and open symbols denote densities com-puted using integration time steps of 1É0 and 0É5 fsrespectively. For convenience, the experimental densitiesat saturation pressure43 are indicated by the full lines.Thus it is observed that at temperatures far from thecritical temperature (e.g. time step e†ectsT [ Tc [ 100),are indeed minor, whereas as the critical temperature isapproached, sensitivity to the time step becomes quiteevident.

Fig. 1. Calculated and experimental43 density data fordimethyl and diethyl ether. Curves denote experimental dataat saturation pressure. Circles and triangles were obtainedfrom 100 ps NPT simulations at zero pressure, with fullsymbols obtained for 1 fs integration steps and open symbolsfor 0É5 fs time steps. Crosses are from simulations at experi-mental saturated vapour pressure and using 0É5 fs time steps.

When comparing calculated with experimental data,attention also needs to be drawn to the precise nature ofthe two types of data to avoid the possibility of drawingerroneous conclusions. SpeciÐcally, it should be notedthat as the critical temperature is approached, the di†er-ence between the density of a hypothetical zero-pressureliquid state and the actual liquid under saturation pres-sure can be quite pronounced. Hence comparisons areonly rigorously valid for simulation data generatedunder saturation pressure (strictly speaking, this state-ment holds at all temperatures, though the distinction isof no consequence far from To illustrate this e†ect,Tc).simulations were also performed on dimethyl anddiethyl ether close to the critical temperature andsubject to saturation pressure (c. 4É4 and 2É9 MParespectively for dimethyl and diethyl ether at 393 and453 K). Here the time step was set to 0É5 fs and thecut-o† increased further to The resulting calcu-14É5 Ó.lated densities, indicated by the crosses in Fig. 1, showquite good agreement with the experimental data,though in view of the sensitivity to pressure in thisregion, a more rigorous systematic study of all potentialsources of error is probably in order. Before concluding,it is worth remarking that issues of integration time stepand the distinction between zero and saturation pres-sure are not expected to be relevant in simulations ofoligomers and polymers, since temperatures experiencedduring study and use are far from the critical point.

Turning attention to the behaviour of other relevantsmall-molecule liquids, Fig. 2 illustrates the zero-pressure volumeÈtemperature behaviour of ethanol overthe extended temperature range 223È473 K, obtainedfrom NPT simulations with 1É0 fs integration time steps.Evidently, agreement between the zero-pressure simula-



Fig. 2. Calculated and experimental43 density data forethanol. The curve denotes experimental data at saturationpressure. Squares are from 100 ps NPT simulations at zero

pressure and using 1É0 fs integration time steps.

tion and the experimental data at saturation pressure isagain excellent, with errors less than 1% from 223 to423 K. The apparent error increases to c. 4% at 473 K,though in view of the importance of the time step andapplied pressure in the vicinity of (516É3 K), this dis-Tccrepancy is probably not signiÐcant.

Data for actual PEO oligomers are illustrated in Figs3È5, which depict speciÐc volume behaviour as a func-tion of both temperature (303È473 K) and pressure (0È180 MPa). Figure 3 pertains to simulations ofheptamers of PEO terminated at both ends withhydroxyl groups (i.e. PEG with a molar mass of326É4 g mol~1). The smooth curves in the Ðgure rep-resent experimental data for a commercial PEG with

and The pointM1 n\ 302 g mol~1 M1 w/M1 n \ 1É18.44marked by the cross represents an additional indepen-

Fig. 3. Calculated and experimental44 speciÐc volume curvesfor Experimental curves refer to 0É1 andHO(CH2CH2O)7OH.180 MPa. Full and open squares denote simulations at zeropressure with 1É0 and 0É5 fs integration time steps respectively.Triangles represent simulations at 180 MPa. The cross denotes

an additional experimental datum at 293 K.45a

Fig. 4. Calculated and experimental44 speciÐc volume curvesfor with conditions analogous to cor-HO(CH2CH2O)14OH,

responding data in Fig. 3.

dent experimental data point at 293 K.45a Focusing onthe data points from the simulations, good agreement isfound for all simulations at 180 MPa and for simula-tions below 373 K at zero pressure. At 423 and 473 K,however, some statistically signiÐcant di†erences, of c.1É5%È2É5%, exist between calculation and experiment.To conÐrm that these di†erences do not result fromintegration-time-step-related artefacts, a set of simula-tions was also performed using 0É5 fs integration timesteps. The resulting points (open squares) do not di†ersigniÐcantly from the standard simulations. Anotherpoint worth noting is that in similar simulations on aPEG containing 14 repeat units (with a molar mass of634É8 g mol~1), good agreement is found at all tem-peratures and pressures with experimental data on acommercial PEG sample with andM1 n\ 692 g mol~1

as illustrated in Fig. 4. Thus the causeM1 w/M1 n \ 1É06,44of the discrepancy observed with the lower-molar-massmaterial at zero pressure remains obscure. Finally, it

Fig. 5. Calculated and experimental44 speciÐc volume curvesfor squares, P\ 0 MPa; circles,HO(CH2CH2O)7OCH3 :P\ 60 MPa; diamonds, P\ 120 MPa; triangles,

P\ 180 MPa.



can be seen in Fig. 5 that the same good agreement, thistime at pressures of 0, 60, 120 and 180 MPa, is alsoobtained for a PEG sample with seven repeat units, butwith one of the terminal hydroxyls replaced by amethoxy group, giving a molar mass of 354É4 g mol~1.Thus the 3%È4% di†erence resulting from replacementof a hydroxyl group44 is correctly reproduced.

The calculated chain length dependence of density isillustrated in Fig. 6, plotted in the form of density as afunction of reciprocal molar mass. In this study, calcu-lations were performed for chains containing from oneto 20 repeat units, with methoxy groups capping eachend. The data at 373É2 K were supplemented by a pointobtained for a 250-mer, corresponding to a molar massof c. 11 000 g mol~1. In contrast with the simulations ofthe shorter chains, this result represents an average ofresults from NPT simulations on Ðve independentlygenerated conÐgurations. The dotted lines through thedata points represent least-squares best Ðts to the data.SpeciÐcally, the data at 303 and 373 K can be represent-ed by the relationships

o303 \ 1É1221 [ 23É67/M

o373 \ 1É0693 [ 26É32/M

where M denotes the molar mass and o the density.Comparison with high-molar-mass experimental resultshas been performed using data of Zoller and Walsh44reported for a commercial PEO sample of molar massc. 105. The results at 303 and 373 K are illustrated bythe horizontal broken lines in Fig. 6. The point at373É2 K was obtained directly by interpolation betweenthe values measured experimentally at 367É4 and375É1 K. Since the high-molar-mass material is semi-crystalline at 303 K, however (since theTmB 341 K),density of the non-crystalline PEO at this temperature

Fig. 6. Calculated density versus reciprocal chain length forfor 1\ x \ 250 : full circles, 373 K;H3CO(CH2CH2O)

xOCH3

open circles, 303 K. Dotted lines denote linear least-squaresÐts. Broken horizontal lines denote experimental values44 for

PEO of molar mass c. 105.

was obtained by Ðtting the experimentally reporteddata44 between 343 and 473 K to a function quadraticin temperature. The resulting expression for the densityof high-molar-mass PEO as a function of temperature is

o(T ) \ 1É1440 [ 7É039 ] 10~4(T [ 273É2)

[ 2É794 ] 10~7(T [ 273É2)2

In conclusion, it appears from Fig. 6 that the densitiesfrom the oligomer simulations can be well representedas linear functions of reciprocal molar mass, whichupon extrapolation give good agreement with experi-mental densities of high polymer. It is notable also thatthe data point for the 250-mer lies close to the best-Ðtline in spite of the fact that the data were averaged overonly a relatively small number of independent conÐgu-rations.

3.3 Cohesive properties

Cohesive energies, in the form of solubility parameters,were examined for dimethyl ether and ethanol as func-tions of temperature and for selected PEG oligomers at298É2 KÈthe latter temperature being chosen in view ofthe large compilations of experimental (and correlation-based) data reported at this temperature.46 To enablecalculations to be extended to within 10È20 K of thecritical temperature without introducing artefactsassociated with factors such as time-step-related densityerrors discussed in the preceding section, all simulationswere based on analysis of NVT simulations performedat experimentally reported densities.43,46 Typically, ÐnalconÐgurations obtained in the earlier NPT studies weretaken and affinely scaled to the appropriate density.This was followed by 110 ps molecular dynamics simu-lations, with data collected every 5 ps for the last 100 psof the run, and with the last 75 ps used in the actualanalysis. Cohesive energies were calculated as describedpreviously as the ensemble average of the intermolecu-lar non-bond energy.

Figure 7 depicts typical behaviour of the solubilityparameter d versus temperature for ethanol between 223and 473 K and for dimethyl ether from 303 to 393 K.The full curves represent values deduced from experi-mental heat-of-vaporization data using the relation

d2\ oM

(*Hl [ RT )

where o denotes the density, M the molar mass andthe heat of vaporization. Note that, strictly speak-*Hl

ing, the heat-of-vaporization data should be correctedfor factors such as gas phase dimerization prior to com-parison with the simulation results. While such dimer-ization does occur for alcohols, the overall e†ect issmall. However, in the absence of all other factors theconsequence of this e†ect would be to make computedvalues of the solubility parameter slightly larger than



Fig. 7. Comparison of solubility parameters of dimethyl etherand ethanol calculated during NVT simulations, with data

estimated from experimental heat-of-vaporization data.43

those estimated from experimental heat-of-vaporizationdata. Notwithstanding these minor considerations,agreement between calculation and experiment is goodfor temperatures well below the critical temperatures.Apparent deterioration occurs at higher temperatures,though it would be unwise to draw deÐnitive conclu-sions regarding any force Ðeld inadequacies in theabsence of a thorough study of the accuracy of this andall other pertinent experimental data. This is particu-larly true for solubility parameters deduced from thedimethyl ether data, which Ref. 43 indicates are*Hlbased on minimal measurements combined with appli-cation of a “standardÏ estimation method for all othertemperatures. Accordingly, at this stage it is simply con-cluded that the simulation data adequately reproducethe experimentally deduced c. 50% decrease in the solu-bility parameter.

Predictions of the room-temperature solubilityparameters for selected PEG oligomers are listed inTable 8, where, in the case of ethylene glycol, diethyleneglycol and triethylene glycol, comparisons are madewith sets of “experimentalÏ values obtained from di†er-ent sources. For the calculated data the numbers givenin parentheses indicate the standard error in the mean

of the solubility parameter calculation. The Ðrst set ofexperimental data was deduced from compiled values ofheats of vaporization43 as described in the precedingparagraph. To give an indication of the range of liter-ature values, two additional sets of data, tabulated inthe monograph by Barton46 and based on extensivecompilations by Hansen47 and Hoy,48 are alsoincluded. While overall comparisons with literaturevalues show reasonable agreement, it is also notablethat there are signiÐcant disagreements between di†er-ent sources. Since there may be several explanations forthese discrepancies between literature values (e.g. thesource and accuracy of the raw experimental data, thor-oughness of the analysis in terms of elimination offactors such as gas phase association, etc.), detailed dis-cussion of the accuracy of the predictions will requirefurther analysis of the original data. One aspect which isworth pursuing, however, concerns estimation of thesolubility parameter of high-molar-mass PEGs. In thisregard it can easily be demonstrated that plotting thedata of Table 8 in the form of the square of the solu-bility parameter (i.e. the cohesive energy density) versusreciprocal molar mass results in an almost perfectlylinear set of points, which can be extrapolated to esti-mate d for high polymer. The resulting straight line canbe represented by the equation

d2\ 418É8 ^ 8É4 ] 50 919^ 935M

which yields an estimate of d \ 20É45 for high-molar-mass polymer. This can be compared with the value of20É2 ^ 0É2 obtained for PEO using the technique ofinverse gas chromatography.45b

4 POLYMERIZATION SIMULATIONS

4.1 Overview and simulation approach

The extensive body of work on urethane networks thatStepto and his colleagues have published has includeddirect studies of cyclization reactions in model systems.

TABLE 8. Comparison of solubility parameters, calculated from NVT simulations at 298·2 K

and experimental densities, with values deduced from experimental data

Molar dcalc

dexp

dexp

dexp

mass (MPa0Õ5) (MPa0Õ5) (MPa0Õ5) (MPa0Õ5)(g molÉ1) Ref. 43 Refs 46, 47 Refs 46, 48

Ethylene glycol 62·07 35·13 (0·05) 34·2 32·9 34·9

Diethylene glycol 106·12 29·97 (0·03) 30·9 29·9 29·1

Triethylene glycol 150·18 27·80 (0·04) 26·3 27·5 29·1

Heptaethylene glycol 326·39 23·99 (0·04) — — —

Tetradecaethylene glycol 634·76 22·11 (0·03) — — —



The early work of Stepto and Waywell20 on intramole-cular reactions in linear polymerizations of PEG withhexamethylene diisocyanate (HDI) was later expandedupon by Stanford and Stepto49 in their work on cycliza-tion in reactions of polyoxypropylene (POP) triols withHDI. In the studies of Stepto and Waywell, titrationswere performed to determine the concentration of unre-acted isocyanate groups and cryoscopic measurementswere conducted to determine number-average molarmasses. By knowing both the extent of reaction fromtitrations and molar masses from the colligative mea-surements, Stepto and Waywell were able to determinethe fraction of reactions that are intramolecular. Anintramolecular reaction forms a cycle, and inasmuch aslarge cycles are elastically e†ective but small cycles areelastically ine†ective, it is very important to understandthese structures in relation to high elasticity. SteptoÏswork20,49 is the most direct that has ever beenpublished on this question.

A major emphasis of SteptoÏs work on cyclization hasbeen to determine the departures from FloryÈStockmayer statistics50 that result from cyclization. Toamplify the departures from these statistics, applicableonly to acyclic reactions, the polymerizing system maybe diluted. It is intuitively clear that cycles are formedin increasing proportion as the polymerizing system isdiluted ; this phenomenon is used as a ploy by syntheticchemists when conducting cyclization reactions. Steptoand Waywell studied “linearÏ urethane polymerization ata variety of concentrations from the bulk all the waydown to a 1% solution. They showed that cyclizationindeed becomes more important as dilution increases,and approaches, but does not reach, the limit where allreactions form cycles.

In our continuing e†orts to improve and validate theMonte Carlo network formation code that we havedeveloped,19 we have undertaken simulations thatduplicate the conditions of the Stepto and Waywellexperiments. The experiments provide a set of resultsthat promise a rigorous test of simulation methods andare particularly pertinent for the Monte Carlo algo-rithm. Previous work with this algorithm51 on thePOP] HDI system investigated by Stanford andStepto gave a generally satisfactory account of theirresults, but discrepancies were noted for dilute reactionconditions, and these could not be completely amelio-rated by manipulating the simulation variables. It wasand still is apparent that one can increase the probabil-ity for ring formation by decreasing the chain length ofthe cyclizing species. E†orts to exploit this fact byskewing the molar mass distribution of the triol armswere not successful in rectifying the discrepancybetween experiment and theory.

At the time that the simulation results on theStanfordÈStepto experiments were presented, the algo-rithm was criticized by Stepto for not including themovement of reacting molecules onto one another as

bonds were being formed. The algorithm uses a proxim-ity bonding scheme to mimic the spatial distribution ofreacting groups in the polymerizing mixture, but mol-ecules were not being moved during the bond formationprocess. This is no shortcoming for high concentrationsof reactive groups, where movement of molecules can beshown to have no perceptible e†ect on the results. Infact, the algorithm gave an exceedingly good account ofthe StanfordÈStepto experiments on bulk and concen-trated systems virtually independent of di†erent, butrational, simulation parameters. However, discrepanciesfor dilute systems could not be satisfactorily resolved, asnoted above.

Because molecules were not being moved, the accusa-tion was that the algorithm was not properly represent-ing the conditional probability that the formation of acycle in the second reaction of an HDI molecule isa†ected by the fact that the Ðrst end has reacted. Ine†ect, the algorithm was treating the reactions at thetwo ends as independent events. It is known from theJacobsonÈStockmayer ringÈchain equilibrium theory52that the probability for cyclization in dilute solution isdependent on the (negative power of the) chain length.32Formation of the Ðrst bond to an HDI molecule e†ec-tively increases the chain length, thereby diminishingsomewhat the probability for cyclization (at least rela-tive to a very short difunctional monomer) according tothe JacobsonÈStockmayer theory. On the other hand,by forming the Ðrst bond between a polyether end andone of the NCO groups of the HDI, the local concen-tration of unreacted isocyanate available to theopposing chain end is increased, thereby increasing theprobability for cyclization. It is not so obvious whiche†ect will dominate at which concentrations, and that iswhy we have been cautious to include this e†ect in thealgorithm. Furthermore, as noted above, so long aspractical applications were primarily to concentratedand bulk systems, there has not been a strong need tomake this modiÐcation to the algorithm.

E†orts to explore the applicability of the networksimulation code to other types of polymerizations, par-ticularly to “linearÏ polymerizations, have caused us toreturn to an analysis of the SteptoÈWaywell data to seehow well the “stationary reactantsÏ method wouldaccount for the data. What was found, in calculationsnot reported here, is that discrepancies loomed largerthe higher was the dilution, reminiscent of the results ofShy and Eichinger51 on the StanfordÈStepto data. Inthe case of the SteptoÈWaywell results there is no ambi-guity in molar masses and MWD to manipulate toimprove agreement. The discrepancy clearly revealed ashortcoming of the method.

We have since modiÐed the code to move moleculesas they react, and now can give a very nice account ofthe SteptoÈWaywell results, with no free parameters.This should help to ameliorate some of the misgivingsabout the Monte Carlo method that have appeared in



the literature. (In particular, the results of Rolfes andStepto53 were obtained in an e†ort to simulatebranched molecules, which the code was not equippedto handle at the time of that work. Their results arehighly dependent on the details of the method used forthe simulation and on the interpretation of the reportedoutput.)

The method of movement is as follows. Molecules aremoved to within their proper bonding distance alongthe vector joining the reacting groups by relative trans-lations of the reacting pairs that are in inverse propor-tion to the molar masses of the molecules to which theyare attached. This particular weighting for the move-ment is a compromise ; for low-molar-mass compounds,e.g. crosslinkers, there is no general rule for the depen-dence of di†usion coefficient on mass, but above a lowerlimit the di†usion coefficient should depend upon thehydrodynamic radius. For a compact molecule thisradius varies as M1@3. For a Ñexible polymer the di†u-sion coefficient varies as M~1@2 for dilute solutions butgoes as M~2 in the reptation regime.54 Rather thanattempt a very complicated rule to mimic this behav-iour over a range of compositions and extents of cure ina crosslinking system, we have opted to implement thesimple “averageÏ behaviour assumed above. This seemsto work quite well. The co-ordinates are updated atevery increment in the bonding distance, not after everyreaction. By updating in batch mode, there is a verymodest decline in the performance of the algorithm. Ifthe reaction is intramolecular, no movement takesplace.

4.2 Chain configurations

ConÐgurations of both the PEG and HDI were gener-ated with the rotational metropolis Monte Carlo(RMMC) method of Honeycutt,55 using the force Ðelddescribed above. For PEG the measured characteristicratio was reproduced by including inter-Cn\ 5É256actions for all atoms separated by at least three bondsbut less than six bonds. Electrostatic interactions wereincluded in the energy calculations with use of chargegroups, and the dielectric constant was set to 1É4. (As inthe case of the bulk simulations described in Sections 2and 3, the atoms in neutral groups are either allincluded or all excluded from the electrostatic calcu-lation, depending on whether the group switchingatoms are separated by distances less than the cut-o†.)The bulk dielectric constant for PEG is about 2É8 ; useof a smaller value for near-space interactions is consis-tent with the local environment being somewherebetween the vacuum and the bulk.

The RMMC-generated populations of rotationalstates using these modelling parameters for a pentamerof dimethyl ethylene glycol at 313É2 K are shown in Fig.8. (This corresponds to the molar mass used in theSteptoÈWaywell experiments. For the present dis-

Fig. 8. Probability distributions of dihedral angles inat 313É2 K. The full curve denotesH3CO(CH2CH2O)5OCH3

OCCO populations ; the dotted curve denotes the overall dis-tribution. Curves were obtained from simulations on “unper-turbedÏ isolated chains subject only to local intrachain

interactions.

cussion of chain statistics, results are reported on thedimethyl derivative rather than the glycol.) It can benoted that the force Ðeld captures the gauche e†ect, asseen in the distribution of torsion angles about the CCbonds (full curve). The distribution of all rotationalstates, for both the CC bonds and the two CO bondsper repeat unit, is shown in this Ðgure as the dottedcurve. Clearly the CO bonds prefer the trans conforma-tion. In this Ðgure the population of trans CC bonds isabout 23%. Values ranging from about 20% to 31%were obtained for the population of trans CC bonds,depending upon whether the starting structure was fullyminimized or was selected from one instant of adynamics trajectory. The sensitivity of rotational popu-lations to the details of bond lengths and bond anglesand their distributions has been noted previously.55(For the bulk simulations reported above, the popu-lations of trans CC bonds ranged from 13% to 22%over a 170 K range of temperatures for triethyleneglycol, while the population of trans CO bonds variedfrom 64% to 71%. No clear dependence of populationson temperature outside the statistical uncertainty couldbe inferred from these results.) The point is that, regard-less of nuances, the force Ðeld clearly predicts that thetwo gauche states for the CC bond are far preferredover the trans.

ConÐgurations for the dimethyl pentamer that weregenerated at 313É2 K to analyse the rotational popu-lations described in the preceding paragraph also gavethe end-to-end distance distribution shown in Fig. 9.The non-Gaussian behaviour is clear ; the end-to-endvector distribution shown in Fig. 10 also illustrates thispoint. To construct the last three Ðgures, 106 MonteCarlo moves were generated, with conÐgurations beingsaved every 10 000 moves. The noise in the curves is the



Fig. 9. End-to-end distance distribution W (r) obtained insimulations of H3CO(CH2CH2O)5OCH3 .

result of a small sample size, but the conÐgurations arewell equilibrated. One signiÐcant advantage of theMonte Carlo method is that “realÏ chain conÐgurationscan be used in the simulations. For this purpose thechain conÐgurations used for the network simulationswere drawn from 1000 conÐgurations of a 200-mer thatwere generated by the RMMC method at 343É2 K, theexperimental temperature. Otherwise, the same param-eter set was used as for the 313É2 K run. Since theMonte Carlo network algorithm uses a distribution ofmolar masses when called upon to do so, it Ðrst gener-ates a chain length and then looks at random for asequence having that length in the supplied conÐgu-ration Ðle generated by the RIS method of choice.

Fig. 10. End-to-end vector distribution P(r) obtained in simu-lations of H3CO(CH2CH2O)5OCH3 .

Finding such a sequence of bonds, it accepts the atom-to-atom distance that is needed to construct the co-ordinates for the reactive groups that are currentlybeing placed in the reaction box. ConÐgurations forHDI were similarly generated, but of course HDI is apure substance and the MWD is trivial.

4.3 Discussion of polymerization simulation results

While Stepto and Waywell did not specify an MWD fortheir PEG, we have assumed a Gaussian distribution ofmolar masses with andM1 n \ 226 g mol~1 M1 w \

This is a tight distribution which we235 g mol~1.believe is a reasonable representation for the breadth ofmolar masses that one might expect in a system likethis. Results would not be signiÐcantly di†erent if thePEG were to be treated as a pure, monomolecular sub-stance.

Experimental results for cyclization in this irrevers-ible “linearÏ polymerization reaction between PEG diolsand HDI were kindly supplied by Professor Stepto fromWaywellÏs thesis.57 The data, drawn directly from thethesis, are shown as the plotted points in Fig. 11. (Asmall di†erence between Fig. 11 and the publishedÐgure20 may be the result of a plotting error in thatpaper.) The curves are from simulations, with no adjust-able parameters. The overall quality of the Ðts is quitegood. Moreover, even closer agreement with experiment

Fig. 11. Number fractions of rings obtained in HDI/PEGpolymerizations in benzene at di†erent concentrations. Curvesare calculated from Monte Carlo polymerization simulations ;individual points are deduced from experimental data.20,57Monomer weight fractions are as follows : bulk ; 20%;K, …,

10%; 5%; 1%.), +, L,



is obtained with an RIS model for PEG that givesslightly smaller The calculated curves are theC= .average results of four conÐgurations for each concen-tration, with each conÐguration containing approx-imately 65 000 reactive groups. In all respects thesimulation conditions mimic the conditions of theexperiments as closely as possible.

Two things are to be noted in the comparison ofsimulations and experiments. In the Ðrst instance, simu-lations overestimate the extent of cyclization in the bulkand at the next highest concentration, as can be seen byinspection of Fig. 11. This is observed both with andwithout movement of the molecules, so there is nochange that can be made in the simulations to rectifythe discrepancy. Secondly, the simulations tends tounderestimate the extent of cyclization at high dilutions.We think that both observations can be explained.

Stepto and Waywell analysed their results with theassumption that there is no catenation. The simulationmethod does not model catenation either. Could theexperiments be a†ected by catenation? The answer isyes. The SteptoÈWaywell20 equation relating toM1 nI/X, the number fraction of rings (their notation), is

IX

\ 1 [ 2(1 [ p)M1 nMa ] Mb

where p is the extent of reaction and the sumMa ] Mbof the molar masses of the reactants. (This equation is avariant of that of Stepto and Waywell for balanced stoi-chiometry.) Now, if there is catenation, is largerM1 nthan it would be in its absence, resulting in a smallercalculated fraction of rings. This is just the directionneeded to account for the discrepancy observed in Fig.11.

We now have enough conÐdence in the algorithm tosuggest that the di†erence between simulations andexperiments be used to determine the extent of cate-nation. E†orts to analyse the data in this manner revealthat very-high-accuracy experiments will be required tomake the exercise meaningful. Given the inherent limi-tations in colligative property measurements, it isunlikely that there can be much improvement with useof the SteptoÈWaywell protocol.

It is suggested that the undershoot at the highestdilution could be the result of some experimental diffi-culties. Looking at just the upper set of experimentaldata in Fig. 11, it appears that a smooth curve drawnthrough the data would not intersect the upper right-hand corner of the plot, as it must. (It is impossible forany point to lie above the line on this plot. TheNr \ plast datum lies practically on this line.) It seems likelythat titrations are more difficult the more dilute is thesolution, and an endpoint in a titration would tend tobe overshot, because the colour change is harder to seethe more dilute is the solution. An error of this sort is inthe direction needed to account for the discrepancy.Regardless of these speculations, in the absence of error

bars on the experimental data at high dilutions it is dif-Ðcult to make deÐnitive conclusions about the qualityof Ðt between simulations and this particular set of data.

5 CONCLUSIONS

The present studies have reported on the developmentof a force Ðeld suitable for making quantitative predic-tions of condensed phase properties of poly(ethyleneoxides) and related alcohols and ethers, and on anextension of a Monte Carlo approach for simulation ofpolymerization and network formation. Two speciÐcapplications, involving prediction of thermophysicaland cohesive properties of PEO oligomers and predic-tion of the extent of ring formation in “linearÏ,polyurethane-forming, step growth polymerizations,using chain conÐgurations generated via the force Ðeld,have been studied in some detail.

A number of conclusions have emerged from theparametrization and atomistic validation studies.Firstly, during the parametrization studies, densitiesobtained from preliminary NPT MD simulations andinspection of the ab initio charge distribution in ethersand alcohols led to recognition of the need to treatcarbon atoms alpha to the oxygens in these moleculesdi†erently from normal aliphatic carbons. While theresulting di†erences in carbon non-bond parameters arerelatively small (e.g. c. 10% change in well depth, \1%in performance of the force Ðeld is measurablyr0),improved by making these changes. This aspect of theparametrization does not appear to have been discussedpreviously.

Secondly, on applying the force Ðeld to atomistic pre-dictions of PVT behaviour, it was found that experi-mental behaviour of liquids and oligomers canfrequently be accurately reproduced (D1%) over broadranges of temperature and pressure, even though thenon-bond parametrization was only optimized forambient conditions. For small-molecule liquids, agree-ment with experiment remains quite good at tem-peratures surprisingly close to the critical temperature,although it was shown that under these conditionscareful steps must be taken to eliminate artefacts associ-ated with simulation parameters such as MD integra-tion time step. Accompanying investigations of thepossibility of using oligomer data to predict the densityof high-molar-mass polymer, which owing to slowrelaxation behaviour would be expected to be difficultto study using traditional MD techniques, proved to bee†ective. Finally, although most of the focus in thiswork has been on liquid and amorphous states, it wasfound that crystal structure data can also be predictedwith an accuracy comparable with that typical forliquid systems.

Rigorous comparisons of predicted solubility param-eters with experiment were found to be more difficult to



make than the density/speciÐc volume data, mostly inview of the uncertain nature of the origin and treatmentof the original experimental data. In this regard, com-monly used sources of so-called experimental datasometimes di†ered signiÐcantly by up to c. 5% (in com-parison, the reproducibility of the simulation results,based on two standard errors in the mean, is c. 0É3%È0É4%). However, where multiple sources of experimen-tal data exist, the simulation data were found to lieeither at or between the experimental extreme values. Inaddition, it was gratifying to Ðnd that extrapolation ofexperimental data as a function of (reciprocal) molarmass leads to close agreement with reported solubilityparameter data for polymeric material.

Lastly, turning to the simulation of the topology ofpolymerizing systems, it can be concluded that a verygood account of experimental cyclization data can behad with simulations, and this without the use of anyfree parameters. The method is truly predictive. Andperhaps most important from the practical sense is thatthe simulation method provides a means to determinethe extent of cyclization in irreversible step growthpolymerizations, thus augmenting JacobsonÈStockmayer theory for reversible systems. The resultsclearly show that as the extent of reaction approachesone, all linear chains are converted into rings. This hasprofound implications for polymer processing.

ACKNOWLEDGEMENTS

We are indebted to Peter Sher for his exceptional con-tributions to the Monte Carlo network simulation algo-rithm. The polymerization simulation work wassupported by the Office of Naval Research, contractN00014-95-C-0156. The PVT studies were performed asa component of ARPA contract F30602-95-2-0007/C-5-2779/AO C277. We wish to thank IBM Corporationand ARPA for providing computational resources forthe PVT calculations.

DEDICATION

This paper is dedicated to our friend, colleague andmentor, Robert F. T. Stepto, on the occasion on his60th birthday.

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APPENDIX

DeÐnitions of atom types and equivalences for alcohols and ethers

Type DeÐnition Non-bond Bond Angle Torsion Out-of-plane

c4 Carbon, sp3, 4 bonds c4 c4 c4 c4 c4c3a Aromatic, 3 bonds c3a c3a c3a c3a c3ac4o Alpha carbon c4o c4 c4 c4 c4o2h Hydroxyl oxygen o2h o2h o2 o2 o2o2e Ether oxygen o2e o2e o2 o2 o2h1 Non-polar hydrogen h1 h1 h1 h1 h1h1o Strongly polar hydrogen h1o h1 h1 h1 h1

Parameters

(Units : bonds lengths in angles in grad, except in deg ; energies in kcal mol~1.)Ó; t0

Bond increments (in electrons)

c3a o2e 0É042 0c3a o2h 0É042 0c4 o2e 0É160 0c4 o2h 0É160 0h1 o2 0É420 0h1 o2h 0É420 0

Quartic bond

i j r0 k2 k3 k4c3a o2 1É376 8 428É879 8 [738É235 1 114É965 5c3a o2e 1É376 8 428É879 8 [738É235 1 1 114É965 5c3a o2h 1É376 8 428É879 8 [738É235 1 1 114É965 5c4 o2e 1É420 0 400É395 4 [835É195 1 1 313É014 2c4 o2h 1É420 0 400É395 4 [835É195 1 1 313É014 2h1 o2h 0É949 4 540É363 3 [1 311É866 3 2 132É444 6



Quartic angle

i j k t0 k2 k3 k4c3a c3a o2 123É420 0 73É678 1 [21É678 7 0É000 0c4 c4 o2 111É270 0 54É538 1 [8É364 2 [13É083 8h1 c4 o2 108É728 0 58É544 6 [10É808 8 [12É400 6c3a o2 c4 102É969 5 38É973 9 [6É259 5 [8É171 0c3a o2 h1 108É190 0 53É125 0 [8É501 6 0É000 0c4 o2 c4 104É500 0 35É745 4 [10É006 7 [6É272 9c4 o2 h1 105É800 0 52É706 1 [12É109 0 [9É868 1

T orsion

i j k l k1 k2 k3c3a c3a c3a o2 0É000 0 4É849 8 0É000 0h1 c3a c3a o2 0É000 0 1É723 4 0É000 0c3a c3a o2 c4 0É000 0 1É500 0 0É000 0c3a c3a o2 h1 [0É690 0 0É509 7 0É009 5c4 c4 c4 o2 0É713 7 0É266 0 [0É254 5h1 c4 c4 o2 [0É143 5 0É253 0 [0É090 5o2 c4 c4 o2 1É100 0 [0É050 0 [0É144 1c4 c4 o2 c4 [0É400 0 [0É402 8 [0É245 0c4 c4 o2 h1 [0É673 2 [0É477 8 [0É167 0h1 c4 o2 c3a 0É951 3 0É115 5 0É072 0h1 c4 o2 c4 0É530 2 0É000 0 [0É396 6h1 c4 o2 h1 0É186 3 [0É433 8 [0É212 1

W ilson out-of-plane

i j k l k2 a0c3a c3a c3a o2 13É042 1 0É000 0

Non-bond (9È6)

i r0 e0c4o 3É815 0É068h1o 1É087 0É008o2 3É300 0É080o2e 3É300 0É120o2h 3É580 0É096

BondÈbond cross

i j k kÈij/jk

c3a c3a o2 48É475 4h1 c3a o2 4É580 0c4 c4 o2 11É431 8h1 c4 o2 23É197 9o2 c4 o2 8É298 3c3a o2 h1 20É657 7c4 o2 c4 [7É113 1c4 o2 h1 [9É687 9



BondÈbond 1È3 (coupling of bonds separated by a bond)

i j k l kÈij/kl

c3a c3a c3a o2 [2É243 6h1 c3a c3a o2 2É051 7c3a c3a o2 h1 1É159 0

BondÈangle

i j k kÈij/ijk kÈjk/ijk

c3a c3a o2 58É479 0 107É680 6c4 c4 o2 2É686 8 20É403 3h1 c4 o2 4É618 9 55É327 0c3a o2 h1 53É861 4 23É922 4c4 o2 c4 [2É811 2 [2É811 2c4 o2 h1 28É580 0 18É927 7

AngleÈangle

i j k l kÈijk/jkl

c3a c3a c3a o2 0É000 0c3a c3a o2 c3a 0É000 0c4 c4 c4 o2 [0É833 0h1 c4 c4 o2 2É592 6c4 c4 h1 o2 3É917 7h1 c4 h1 o2 2É425 9c4 c4 o2 c4 [3É574 4c4 c4 o2 h1 0É168 9h1 c4 o2 h1 2É128 3

End bondÈtorsion

i j k l ij/ijkl kl/ijkl

k1 k2 k3 k1 k2 k3c3a c3a c3a o2 0É000 0 0É265 5 0É000 0 0É000 0 4É890 5 0É000 0h1 c3a c3a o2 0É000 0 [1É586 7 0É000 0 0É000 0 4É264 1 0É000 0c3a c3a o2 h1 0É900 0 [1É345 6 1É190 0 3É413 2 0É587 3 [0É132 3c4 c4 c4 o2 [0É319 0 0É441 1 [0É717 4 1É153 8 0É840 9 [0É913 8h1 c4 c4 o2 0É968 1 0É955 1 0É043 6 0É590 3 0É666 9 0É858 4o2 c4 c4 o2 1É016 5 0É755 3 [0É460 9c4 c4 o2 c4 [0É245 6 1É051 7 [0É779 5 0É474 1 1É263 5 0É557 6c4 c4 o2 h1 [0É580 0 0É900 4 0É000 0 0É000 0 0É534 3 0É902 5h1 c4 o2 c4 [0É605 4 1É333 9 0É964 8 [0É162 0 0É156 4 [1É140 8h1 c4 o2 h1 [1É755 4 1É314 5 0É226 3 0É249 3 0É680 3 0É000 0



Middle bondÈtorsion

i j k l kÈjk/ijkl

k1 k2 k3c3a c3a c3a o2 0É000 0 4É825 5 0É000 0h1 c3a c3a o2 0É000 0 5É543 2 0É000 0c3a c3a o2 h1 1É158 0 3É269 7 3É513 2c4 c4 c4 o2 [21É884 2 [7É676 4 [0É686 8h1 c4 c4 o2 [16É797 5 [1É229 6 [0É275 0o2 c4 c4 o2 [17É258 5 [3É615 7 [0É836 4c4 c4 o2 c4 [5É928 8 [2É700 7 [0É317 5c4 c4 o2 h1 1É247 2 0É000 0 0É748 5h1 c4 o2 c4 [6É800 7 [4É654 6 [1É410 1h1 c4 o2 h1 0É000 0 0É924 1 [0É588 9

AngleÈtorsion

i j k l ijk/ijkl jkl/ijkl

k1 k2 k3 k1 k2 k3c3a c3a c3a o2 0É000 0 10É015 5 0É000 0 0É000 0 1É740 4 0É000 0h1 c3a c3a o2 0É000 0 1É872 9 0É000 0 0É000 0 2É570 6 0É000 0c3a c3a o2 h1 [5É136 0 [1É012 2 0É000 0 4É685 2 0É023 0 [0É598 0c4 c4 c4 o2 0É562 3 [0É304 1 [0É401 5 0É967 2 [0É756 6 [1É233 1h1 c4 c4 o2 2É366 8 2É492 0 [1É012 2 [0É189 2 0É491 8 0É727 3o2 c4 c4 o2 0É551 1 0É973 7 [0É667 3c4 c4 o2 c4 [2É746 6 1É487 7 [0É895 5 0É567 6 0É945 0 0É070 3c4 c4 o2 h1 [3É590 3 2É522 5 0É488 8 0É872 6 [0É357 7 0É388 8h1 c4 o2 c4 [1É823 4 1É639 3 0É514 4 [0É777 7 0É434 0 [0É665 3h1 c4 o2 h1 [3É406 0 1É639 6 0É073 7 0É000 0 [0É281 0 [0É594 4

AngleÈangleÈtorsion 1

i j k l ijk/jkl/ijkl

c3a c3a c3a o2 [21É024 7h1 c3a c3a o2 4É229 6c3a c3a o2 h1 [4É607 2c4 c4 c4 o2 [29É042 0h1 c4 c4 o2 20É200 6o2 c4 c4 o2 [14É048 4c4 c4 o2 c4 [19É005 9c4 c4 o2 h1 [12É103 8h1 c4 o2 c4 [16É443 8h1 c4 o2 h1 [10É509 3


computer simulations of poly(ethylene oxide): force field, pvt diagram and cyclization behaviour

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