Meander Model of Condensed Polymers BY WOLFGANG R. PECHHOLD AND HANS P. GROSSMANN Abteilung Angewandte Physik, Universitat Ulm, Oberer Eselsberg, D 7900 Ulm, W. Germany Receiued 30th May, 1979 On the basis of the cluster-entropy hypothesis (CEH) the short-range order of polymer melts can be described by clusters of nearly parallel chain segments having similar conformations and holes in between these clusters, This bundle model allows the quantitative discussion of the melt transition as well as of the thermal properties of the melt. Within these bundles the molecules are assumed to fold back and forth in a one-dimensional statistical manner defined by the free energy of a tight chain fold. The bundle diameter, r, and the superfolding of bundles are derived by applying the CEH again.As primary blocks (5-30 nm, depending on the kind of polymer and molecular weight) coupled meander cubes are most probable. These are linked via their cube diagonals which serve as the axis of statistical rotation. Appropriate interaction between adjacent cubes may account for a liquid- crystalline transition, in which the paracrystalline but isotropic grains (0.3-3 ,urn) become nematically ordered. The diameters of these coarse grains (found in most amorphous polymers) depend on their grain boundary interaction. The shear deformations of the coupled cubes explain quantitativeIy the rubber-elastic compliances, JoeN, of uncrosslinked polymer melts. Additionally, the unfolding of layers of meander-cubes into shear bands describes the stress-strain relation of high molecular weight polymer melts.1. INTRODUCTION Our current knowledge about the level of order in amorphous polymers should stimulate further development of competing molecular models, by making their suppositions more precise in order to provide a bridge between their microscopic structure description and the understanding of macroscopic properties, thereby predicting effects which might be proved experimentally. No property or correlation should be excluded from consideration. The model theory presented here is based on the cluster-entropy hypothesis (CEH) and additionally assumes that clusters of nearly parallel chain segments and, therefore, chain bundles are energetically favoured even though some conformations have to be excluded. are improved by taking into account topological aspects and all kinds of energetic and entropic contributions to the Gibbs free energy of the equilibrium superstructure.A more detailed check and derivation of some of the results is given el~ewhere.~ For systems in or not far from equilibrium, entropy is defined in thermodynamic statistics by Boltzmann's formula or via the partition function, both of which have to be described in I'-space. For condensed phase systems a factorization into sub- space coordinates is generally impossible, but is frequently assumed without proof. A projection into a subspace must include all other states as well as those of special interest. To take this into account we use the following hypothesis (CEH, which has still to be proved) : clustering in subspace (e.g., conformational, orientational or describing deformation) of m equivalent elements (e.g., segments, segment-lines or In this paper earlierW .R. PECHHOLD AND H. P . GROSSMANN 59 layers of molecules), each havingf accessible states (including vibrational states), does not reduce entropy as long as rn <f, i.e., multiple occupation in the p-space can be neglected (or the dimensionality of the r-space is unaltered). There are several applications of this hypothesis in various fields, for example phase transitions in paraffins, membranes, liquid crystals, the melt transition of close-packed crystals, and particularly amorphous structures and related properties of condensed polymers and biopolymers. 2. SHORT-RANGE ORDER CONCEPT Whereas perfect polymer crystals consist of parallel chains of only one low energy conformation, high temperature phases exist (e.g.in PE and t-1,4-PBD) that have accumulated more than half of the total entropy and enthalpy of melting, but which still appear crystalline in WAXS. Transition data and thermal properties as well as their pressure dependences can be quantitatively described, assuming a mixture of ener- getically favoured chain conformations. On the basis of CEH this is realized in conformational c l u ~ t e r s . ~ - ~ Taking a rotator (C2H4 in PE) as one statistical element in the chain direction, a one-dimensional cooperativity leads to the chain segment (x 4 rotators), the mean distance between conformational changes [indicated as slight bends in fig. l(a)].These segments (as new elements) now cluster laterally, retaining the full conformational entropy ( z R In 5 per mol rotator) even though the elements in one cluster show the same conformation at any one time. The actual conformation of an element changes with time and cluster position, i.e., by fluctuations. The subsequent melting is described by the cooperative formation of holes accumulated in the centres of numerous dislocations nearly parallel to the chain direc- tions. The quasi-crystalline clusters (of 10-30 segments) are thereby broken into short fibrils [fig. I@)]. The additional entropy of this transition is caused by the variety of differently shaped holes and their distribution. The chain segment is reduced to x2 rotators (in PE melt), representing a length, s, which compares with the chain distance, d.The assumption s z d will therefore be made for melts of other flexible chain polymers. The Gibbs free energy of a chain segment referred to the melting temperature amounts to g , = h,( 1 - T/Tm), h, ( z 4 kcal mol-I for PE) being the heat of melting per segment. 3. ONE-DIMENSIONAL FOLD-STATISTICS I N BUNDLES OF MOLECULES This concept of short-range order favours chain parallelism across longer distances, because mutual penetration of neighbouring chains should give rise to a greater increase in energy (larger voids or chain deformations) than the respective gain of entropy. In addition to segment clusters and hole distribution, any types of defects which leave the chains nearly parallel and which do not require too much energy must be con- sidered.It has been emphasized already in early papers that torsional defects, jogs and folds may be incorporated into a bundle of molecules. Their concentrations can be esti- mated by their Boltzmann factors. Small clusters formed by these defects (e.g., a pair of folds) should not reduce their probability according to CEH. With regard to the overall geometry of a (labelied) macromolecule in the melt, tight chain folds are most important. Let n be the number of chain atoms (or units after which folding can take place) and Agf = gf - nfgs * the free energy of formation * Forming a tight chain fold, e.g., in PE-melt, the main energetic contribution gf deduced from semi empirical potential calculations is 3.9 or 3.2 kcal mol-', depending on whether one starts with a60 MEANDER MODEL of a tight chain fold, the free energy 6G of one-dimensional folding of a long enough chain is 6G = npfAgf - kWpflnpf + (1 - p f ) M -pf)], pf = e-A.g‘/kT/(l + e-Agf/kr) (6G/RT),q = nln(1 - pf) N” -npf per molecule.(3.1) (3.2) (3.3) pf denoting the fold probability. In equilibrium pf becomes and ( C ) FIG. 2.-(u) Labelled folded chain in a straight melt bundle. (6) Space-filling superfolding into 9-fold meander cubes. (c) Lamella, superfolded, of melt-crystallized bundles. To calculate, e.g., the mean square of the radius of gyration {z;) of a (labelled) folded chain within a narrow straight bundle of molecules [fig. 2(a)] one has to consider the mean square of distances zl between pairs i andj if k = j’ - i’ andfk denotes the number of chain folds between units i’ andj’, Z i F Zjt = (- l)fk and its conformational average can be written trans-chain or tgtg-conformations.But within a fold nf % 1 segment (2 rotators) will be effectively immobilized (similar to the crystallized segment). Their free energy of melting must therefore be subtracted. So Agr = gf - g, = gf - h, + h,T/T, becomes approximately proportional to T.FIG. 1.-(a) Stuart model for the high-temperature phase in PE. (6) Stuart model for the PE melt containing holes between small clusters of segments. [To face page 60FIG. 5.--(a) Equilibrium topology of 9-fold coupled meander cubes linked via their cube diagonals. (b) Possible topology of a cube diagonal linkage. [To face page 61W. R .PECHHOLD AND H . P . GROSSMANN 61 Carrying out the twofold sum one arrives at l 2 l2 2Pf 2Pf (3.6) ( z 5 ) x - ( j - i ) - -2 [I - exp[ - 2 p f ( j -i)]. The mean square of the radius of gyration (zf) is related to (z&) by <z:> = (n + w2 2 2 <Zi.j> (3-7) O < i < j < n and becomes (2:) N" { 1 - [ 1 - 2 (1 - i)]} = a(npf). (3.8) 6Pf 2nPf 2npf 2npf 6Pf This approximate formula is valid for n @ 1 and np, > 1. A representative folded chain labels a piece of a bundle with the length A* x ( I ~ ( z : ) ) ' / ~ . (3.9) The chain has shortened by folding to A*/nl of its extended length nl and its effective cross-section contains on average q = nl/A* = (npf/4a)'I2 (3.10) From eqn (3.6) the scattering behaviour of a folded labelled chain in parallel stems.the straight bundle can also be easily derived. 4. SUPERFOLDING OF BUNDLES So far two main questions remain unanswered: (i) how large is the diameter, r, of the bundle and (ii) how far does it extend in one direction. Both problems can be solved simultaneously by using equilibrium statistics and essentially CEH, if one adopts the concept of superfolding of bundles, i.e. the meander model. The basic idea is to gain orientational entropy if bundles change their directions. The most simple topology in which a melt-bundle can tightly superfold closely to fill 3-dimensional space is an arrangement of meander cubes [fig. 2(b)]. To enable these cubes to occupy all 3 accessible respective sites, the meandering bundle is assumed to link them uia their cube diagonals serving as axis of statistical rotation.Besides superfolding and rotation a statistical description must also take into account the possible shear deformations of a meander cube. For the quantitative treatment statistical elements must be defined which are adapted to control superfolding, rotation and deformation, respectively. These elements should be as small as possible but must possess enough accessible states that their clustering does not violate CEH. As such elements we have chosen (fig. 3): (i) the segment-line of the bundle containing rld segments of length s (for superfolding), (ii) the segment-line across the meander cube consisting of (r + x)/d segments (for cube rotation) and (iii) molecule-layers of the cube which are composed of (r + x ) ~ / sd segments (for shear deformation).The excess free energy per segment due to the superfolding of a bundle into a meander cube is where Agfold refers to a half meander layer of r(r + x)/ds segments and is determined by the free energies of formation of one superfold (Ag:), 2r/d chain bends62 MEANDER MODEL the orientational entropy of segment-line clusters which differ in chain direction (fig. 3) Ag; ban in principle be deduced from potential calculations and probably will be about twice the value of Agf (the free energy of formation of a tight chain fold, which actually is the core of the superfold at least for a high degree of polymerization). According to similar immobilization effects * Ag: will also be approximately propor- FIG. 3.-Representative superstructural unit and definition of the statistical elements.tional to the absolute temperature. The contributions Ag,,, and Agdef in eqn (4.1) will be discussed in the next two sections. Depending on the kind of polymer, AgJ kT has values in the range -In 3 to -0.13, whereas 2 Agdef/kT w -9 for the un- deformed melt. By inserting (4.2) into eqn (4.1) Ag,/kT can be written d s r (1 ++o +:) +;( s Agdef/kT l+x,r)+*]. (4.3) This excess Gibbs free energy per segment in a meander must become a minimum if the independent variables r/s and x / r t take up their (most probable) equilibrium values. From follow the equations -f From topological arguments in ref. (3). x / r has been fixed at a value of 2 but coupling of adjacent cubes (fig. 4) allows some fluctuation of x / r around this mean value.W .R . PECHHOLD A N D H . P . GROSSMANN 63 2Agdef/kT] + :iF A8rot and (1 + $ln( 1 + 5) + In :-= $2$ + + xlr kT ' the difference of which yields -= r A g W s (1 + x/r)ln(l + r/x)' In table 1 the mean values of x/r and r/s are calculated for possible energies of superfold formation using in addition the approximations TABLE 1 .-x/r AND r/s VALUES FOR DIFFERENT Ag; gi/kT VALUES < X I 0 1.36 1.76 2.01 2.17 2.29 2.39 <rls> 3.85 4.83 5.76 6.66 7.55 8.44 According to semi-empirical potential calculations of tight folds by Grossmann and Becks who found 3.2-3.9 kcal mol-1 for PE and 4.2 kcal rno1-l for PS, we assume the corresponding free energies of superfolding to lie in the ranges 6 < Agr/RT < 8 and 7 < Ag;/kT < 9 for PE and PS, respectively.The mean bundle diameters (in high molecular weight melts) should probably be 25-30 for PE (d z 4.8 A) and 50-60 A for PS (d x 8.5 A). The equilibrium Gibbs free energy of superfolding becomes [using eqn (4.3) and (4.5) 1 Fluctuations in x/r and r/s should be allowed for but should not violate topology. Relying on CEH these fluctuations can be described by reversible coupling of any two adjacent cubes in suitable relative orientation (x/r-fluctuation, fig. 4) and by changing the contour of the bundle cross-section without varying its area. The latter cooperative change corresponds to a fluctuation of r/s which is the number of chains in a segment-line, but holds the numbers of chains and stems within the bundle con- stant. FIG. 4.- -fluctuation within coupled meander cubes.64 MEANDER MODEL 2 0 - 10 The proposed topology [fig.5(a)] of %fold meander cubes, i.e. ( x l r ) = 2, is in accordance with equilibrium statistics (table 1). Therefore the concept of a statistical rotation of these cubes about one of their cube diagonals which enables the primary blocks to compose an isotropic grain is also justified. Below a certain temperature however, (if the melt can reach it) the grains should become anisotropic: this will be discussed briefly in section 5. Before going on to this point we still have to answer the question whether or not the bundle diameter, r, depends on chain molecular weight. Because x / r = 2 for topological reasons, r/s will change proportionally to the free energy of a superfold according to eqn (4.6).The idea now is that pairs of chain ends (from different chains) will be substituted for the tight chain folds within the cores of the superfolds, thereby strongly reducing Agi. Assuming the lower limit (probably valid for PS) for r in the case of finite chains I - I I I I I I l l l I I l 1 1 1 1 l 1 I I I I 1 1 1 1 r(r + x) r = r , (1 - w), with w = ~ nld (4.8) being the probability for tight fold substitution (approximately equal to the ' geo- metrical concentration c ' of pairs of chain ends per half meander layer) one finds "[(l +i)1'2- l]whereA=- nld c"2 3r$' (4.9) From this formula the molecular weight dependence of the mean side length 3r(M) of a meander cube in PE and PS melts is obtained and shown in fig. 6. The curve lo3 104 105 M 106 FIG.6.-Molecular weight dependence of the meander size 3r and the radius of gyration R, for PE- and PS-melt. (a) R,PE, (b) R,PS, (c) 3rps, (d) 3rPE. I'M Rf x 4(3r)' + ~ - R TIn pr/ M, 11%. dlA r,/A kcal mol-I PS 52 1 . 2 8 . 7 60 4 . 7 PE 14 1 . 2 4 . 8 25 4 . 4W. R. PECHHOLD A N D H . P . GROSSMANN 65 for PS fits (without further assumptions) the SANS maxima from ref. (9), which we interpret as the scattering of a paracrystalline cubic superlattice labelled with deuter- ated chain ends preferentially gathered in the superfold lines. Finally the radius of gyration R,(M) of completely labelled chains in the coupled- cube meander topology must be estimated. From eqn (3.9) we know the mean length, A*, of a straight bundle that is labelled by the tightly folded chain under consideration.This piece now labels its part of the meander superstructure [e.g. the upper bold path in fig. 5(a)] which in the first approx- imation can be considered as a tetragonal block with dimensions 3r x 3r x A. A is connected with A* by a %fold reduction in length and subsequent stacking of meander layers along cube diagonals (4.10) Using the R,-formula for a tetragonal block the squared radius of gyration of a labelled chain within the superstructure becomes A E (A*/9r)rd%= A*d?19 = (~;)~'~2/3. (4.1 1) with u(npf) taken from eqn (3.8). The molecular weight dependence of R, for PE and PS according to eqn (4.1 1) is plotted in fig. 6 together with some SANS data from various authors.10 The scattering behaviour of labelled chains can also be treated approximately in closed form or by computer simulation.This will be discussed in a later paper,ll together with model considerations for paraffin melts, the superfold cores of which are made up by pairs of chain ends only. At the highest molecular weights the q21(q) against q plot should exhibit a maximum near 5/Rg according to the meander model. Maxima have so far been found in the case of strong clustering, which we interpret on a similar basis : if N labelled molecules are jointly packed within a bundle (on thermodynamic or kinetic grounds) they appear as if they were one molecule having a radius of gyration RgaPp FS R, dz Such a relation has been found to be true by Schelten et all2 5. ORIENTATION OF SUPERSTRUCTURAL UNITS-LIQUID CRYSTALLINE POLYMERS To describe the mutual arrangement of adjacent meander cubes and the overall orientation-as far as it exists-within a grain we write down the Gibbs free energy Grot of hindered statistical rotation assuming 3 equivalent orientational positions (k = 1,2,3) around the linking diagonal of each cube and taking into account the various boundary free energies G:, between adjacent cubes (pair approximation).For a system of N cubes (x/r = 2) composed of m = (r + x)~/s d segment-lines con- taining (r + x)/d segments each, GJN becomes c:, denote pair concentrations along direction i of adjacent cubes in k and I orient- ation and p:, their a priori probabilities. For simplicity it has been assumed that rotational symmetry exists around a possible " director " (z-axis), z 3 1/3 being the concentration of cubes with fold directions parallel to it.The CEH factor rn in front66 MEANDER MODEL of the entropy term accounts for the fact that segment-lines are the statistical elements and clustering into cubes does not reduce orientational entropy (only if rn < f). This fact must be kept in mind if one deals, for example, with magnetic birefringence or with depolarized light scattering in the meander model. Introducing GLl/m = gLl, the free energy per segment on the boundary between cubes, the rotational free energy per segment-line Ag,,, = Gr,,/mN can be written Agrot/kT = 241 [gL/kT + In (cil/~L)I i k l + zlnz + (1 - z)ln(l - z) - (1 - z)ln2 + CA'(2 - cfz - c& - Ct,) + Cp'(1 - z - cfex - c;, - CiX - CiY - 24,). (5.2) The last two terms represent the 6 necessary relations among the ckI, the A', p i being Lagrange multipliers.The subsequent minimization referred to ckl and to the order parameter q = 22 - 1 is carried out el~ewhere.~ We simplify further by assuming gmax being the mean excess free energy per segment of a crossed-chain boundary which will be calculated in the near future using semiempirical potentials. (The shaded areas in fig. 7 indicate such energetically unfavoured crossing of chains.) For this simplified case the results of the minimization are a + 2 aAgrot'kT = -2ln2 + 31n(l + E) + 21n 3 + 31n - = 0 (5.4) ar 1 + r l a - From the equilibrium condition (5.4), plotted in fig. 8, the equilibrium values of are determined by the points of intersection with the zero line for any temperature: for T > T,,, q = - 1/3 (Pz = 0), for T < T,,, q increases from 1/3 to 1 at low temper- atures.For T = T,, q = &1/3 are equally probable and the system undergoes a phase transition which corresponds to the nematic/isotropic transition of low molecular weight liquid crystals. The transition temperature T,, follows according to point symmetry of the S-shaped curves Therefore isotropic polymer melts with an excess free energy g,,,., (e.g. of 0.8 kcal mol-l) in the crossed-chain boundaries should change over to a nematic phase at T,, z 400 K, as long as neither crystallization nor glass transition occur at still higher temperatures. Associated with this hypothetical liquid-crystalline transition would be a latent heat, 6AhrOt, which can be calculated from Agr,,/kT as a function of T/T, [plotted in fig.9(a)] viaW. R . PECHHOLD AND H. P. GROSSMANN 67 Z b X I / FIG. 7.-Table of all possible boundaries between adjacent cubes indicating crossed- chain areas. the enthalpy function [fig. 9(b)], which in the example chosen exhibits a discontinuity 6AhrOt x 0.25 kT, z 0.2 kcal mo1-I of segment lines. This would correspond to 13 cal mo1-1 segment or 0.25 ca1g-I in the case of PE and to 10 cal mol-I segment or 0.03 cal g-I for PS, respectively. Notwithstanding these small transition heats, the mere occurrence of a liquid-crystalline transition (to be observed optically or by electron diffraction) in one or other flexible chain polymers will strongly support the meander concept. We believe that there is no fundamental difference in melt- superstructures between stiffer chain polymers (which more often show a liquid crystalline phase) and flexible chain polymers.So far polydiethylsiloxane (PDES),13*14 which is definitely a flexible chain polymer (T, x 140 K, T, x 268 K), has been shown to exhibit a liquid crystalline phase up to 25 K above T,. 6. SHEAR FLUCTUATION AND DEFORMATION OF THE MEANDER SUPERSTRUCTURE The coupled meander cubes may perform shear fluctuations which should be re- stricted to a displacement between adjacent layers of molecules such that their close packing or superfolding is not violated. This topological assumption compares with that of entanglements in the coil model. The two main intra-meander shear modes are visualized in fig.10, for a single cube: the cross-sectional shear yZl (or y12) and68 MEANDER MODEL the intra-bundle shear ygtra (or yztra). They give rise to an ideal paracrystalline appearance of a polymer melt (fig. 1 1 ) . The mutual glide motions between ad- jacent molecule-layers are probably performed by at least one dislocation per layer, the centres of which can be considered as a linear array of segmental holes of energy &h that are much smaller than a missing segment. -1 0 1 FIG. 8.-Rotational free energy derivative to determine equilibrium values of the order parameter. T/T, = (a) 0.8, (b) 0.85, (c) 0.9, (d) 0.95, (e) l.O,( f ) l . l , (g) 1.2. The anisotropic mechanical compliances corresponding to these fluctuations are essentially of entropic nature and compare to the rubber-elasticity JoeN of the topological " network " of coupled meander cubes.A straightforward statistical derivation on the basis of CEH leads to feat- cross-sectional and intra-bundle shear, respectively, x NN l/qT, accord- ing to the above mentioned restriction. These anisotropic compliances (which are hardly influenced by a low degree of chemical crosslinking) can be measured at higher frequencies and small amplitudes, for example on a stretched rubber sheet (fig. 15).16 For larger extension ratios, A, one measures 2 5 3 1 3 1 (or 2&32) w 41 w 4.8 x m2 N-l and S12,, (or S2121) w JI w 3.5 x m2 N-l, the ratio S12,2/S3131 beingW . 1 0.8 0.6 0.4 0.2 PZ 0 R . PECHHOLD A N D H . P . GROSSMANN 69 . 2 .4 .6 .8 .o -0.2 I1I-. 2 0.5 I I i 1.5 5 I lb T / 7, A9mt - kT 0.5 1 1.5 T I Tu FIG. 9.-(a) Equilibrium free energy Ag,,,/kT of hindered cube rotation and the order parameter t] as function of the reduced temperature T/T,.(6) Rotational enthalpy Ahro,/kTu as function of the reduced temperature T/T,. 1.46 experimentally compared with 1.50 from eqn (6.1) and (6.2). For isotropic samples the rubber-elastic compliance JoeN can be calculated using Reuss’ averaging (* * * a) (Agrot/kT)leq; (-) Pz = &(32 - 1) = t ( 3 q + 1). These compliances are therefore a sensitive measure for the size of superstructural units in polymer melts, to be characterized by 3r/d. There are two more possibilities of obtaining information on 3rtd from molecular motion:17 (i) an analysis of the activation curve of the glass-process, the freezing-in of these shear fluctuations. In the meander model thef,,,.(103/T) dependence of the main relaxation measured by G’, G” can be described by the first part of which is the jump frequency of a segment, the latter the probability of finding at least one representative small hole (of energy ch) within each of 3 ( 3 r / ~ i ) ~ d/s70 MEANDER MODEL 3 1 2 '1212 } = [?'('') 1' cross -sectional shear max 2kT s2121 intra intra FIG. 10.-The two main intra-meander shear modes demonstrated on a single meander cube. segment-lines, i. e. of having available enough dislocations to enable shear fluctua- tions: (ii) whereas the G"-maximum is due to the shortest shear fluctuations (or Rouse modes) the displacements of whole layers of molecules contribute most of all to the glass relaxation measured in the compliance J ( u , T).The rubber-elastic shear displacements ( y I 2 or y3J of a coupled meander cube require about 2 ( 3 r / ~ I ) ~ jumps of segments in one direction, i.e. 4(3r/d)4 jumps during diffusional motion. The frequency shift between the G"- and J"-maxima should therefore depend on the size of superstructural units according to f $klf;l;Hx. = 4(3r144. (6.5) In table 2 the 4 experimental and theoretical methads having been used so far for getting information on 3r/d are listed, together with their application to polyiso- butylene (PIB).W . R . PECHHOLD A N D H . P . GROSSMANN 71 FIG. 1 1 .-Paracrystalline meander topology of the polymer melt induced by shear fluctuations. TABLE 2.-uPPER AND LOWER LIMITS FOR FIVE DIFFERENT DETERMINATIONS OF 3r/d lower upper method for determination of 3r/d limit limit meander theory [eqn (4.6)] 10 30 JoeN = 3.0 - - * 3.7 x 21 activation diagram analysis [eqn (6.4)] 15 21 ratio of relaxation frequencies [eqn (6.5)] 18 24 label experiments 15 24 m2 N-I [eqn (6.3)] 19 The theoretical uncertainty arises from the lack of a reliable tight fold energy, which we are about to calculate atomistically.With the aid of dipolar label molecules which deposit (energetically favoured) into the meander folds at low label concentration the number of these folds and hence 3r/d has been resolved from an anomalous ccn- centration dependence of the dielectric relaxation strength.'* In table 3 similar analyses for various polymers are listed which so far are not as complete as for PIB.Additionally included in this table have been lower limiting values for the enthalpy relaxation calculated according to which can be compared with experimental data already known. Beyond these intra-meander shear fluctuations large scale deformations become possible by inter-bundle displacements, i.e., by unfolding of suitably arranged meander cubes.* To this purpose whole layers of meander cubes must cooperate and form one shear band. An intermediate type of superstructure being thereby produced is indicated by the file of sheared cubes in fig. 12. Fig 13(a) shows the stress-strain behaviour 031 (ygter) of an already developed shearband, the probability P(031) of which depends strongly on the rotational free energy per segment-line Ag,,, [fig.9(a)] that is lost in the shear band. In fig. 13(b) the fully drawn curves give the true shear- *A still further reversible extension can be achieved by expanding the tightly folded chains within the bundle, a process which probably has taken place in ultra-highly-drawn fibres.72 MEANDER MODEL FIG. 12.-Shear band type of deformation shown on a single file of meander cubes. stress-shear-strain dependence for superimposed intra- and inter-bundle shear ~ 3 1 = ygtra + /?ygter. Fromt his molecular deformation theory [completely derived in ref. (3)] the stress-strain behaviour of any type of deformation can be evaluated if the meander size and the rotational free energy AgJkT are known for a given polymer. In fig. 13(d) the theoretical curves for uniaxial tension are compared with measure- ments on high molecular weight PE.Fig 13(c) shows the dependence of the mean distance d, between shear bands (fig. 14) on the extension ratio A. dz can probably be determined by measuring the spacings of the line-by-line structure revealed after oxygen ion etching by electron microscopy of stretched p01ymers.l~ These line structures together with an orientational analysis of the anisotropy of mechanical compliances (fig. 15) support strongly the shear band concept. 7. FINAL REMARKS It is beyond the scope of this paper, which should present the current status of the meander model, to go into all the details of explaining magnetic birefringence, de- polarized light scattering or neutron scattering results on the basis of the meander model.It may be noted, however, that further development of the meander model is in progress in order quantitatively to account for important properties like the dependence of rubber-elasticity on chemical crosslinking or the rheological behaviour of polymer melts. Shear bands, for example, may serve as nuclei for the high tem- perature phase transition and hence also for extended chain crystallization. Finally we will try to answer the often-repeated question whether or not theW. R. PECHHOLD AND H . P . GROSSMANN 73 N 5 0.6 b 0.4 0.2 cr) Ocr) 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x p (O31) FIG. 13.-Deformation behaviour of a high molecular PE melt (a) anisotropic stress-strain curves of intra- and inter-bundle shear. x / r = 2, s = d = 4.8 8, r/d = 5.~2::: = 1/.t/ 3, ~2::: = 9, T = 433 K. (i) 031 (Y:!'~'); (ii) aZ1 (yZ1); (iii) a31 (y:?'"'); (b) serial superposition of intra- and inter- bundle shear (fully drawn curves) and concentration of shear bands (dashed curves): -Agr,t/kT = (i) 1.0, (ii) 0.5, (iii) 0.13; (c) mean distance of shear bands as a function of extension ratio [conditions as for (a)] (i) 1 .O, (ii) 0.5, (iii) 0.13 ; (d) uniaxial stress-strain curves for PE melt (the dashed curve has been measured on high molecular weight PE) (i) 1.0, (ii) 0.5, (iii) 0.13. - meander model, which must be considered as a topological framework wherein the macromolecules are free to reptate, may be valid in concentrated or even in semi- dilute solutions.Our proof for its validity comes from a quantitative understanding of the conversion factors of P, + Py crosslinking reaction (in a 1 : 1 mixture) carried out by Vo11mert20 and coworkers: assuming that the bundle diameter, r, and the bundle superfolding are independent of dilution (below 80% of polymer content) a decreasing amount of back-and-forth folded chains will be incorporated within the bundles with decreasing polymer content. To find the probability, P, of adjacent chain stems which may undergo a crosslinking reaction one easily calculates across one lateral direction where c is the polymer concentration and 4 is the mean number of parallel stems per macromolecule, as given in eqn (3.10). This result would only be true in a parallel array of different bundles.Taking into account superfolding, the stems of the same macromolecule (of sufficient high M ) become adjacent to each other and must therefore be excluded from P'. Only every third contact in one lateral direction occurs74 MEANDER MODEL FIG. 14.--Geometry of deformation for uniaxial extension at different draw ratios. FIG. 15.-Five anisotropic compliances of a stretched rubber sheet measured at small amplitudes as a function of the extension ratio at 100 “C, 10 Hz.” +, 41; 0, JL ; x , DI ; 0, DII ; A, JT. 2S313, =2S3232 = 4.8 x m2 N-I, 2S2121 = 2S1z12 = 7.0 x m2 N-l, all other Srlcr ~ 0 .W. R . PECHHOLD A N D H. P. GROSSMANN 75 between different bundles and contributes to the crosslinking probability. Therefore The factor 1/2 accounts for the probability of finding two adjacent molecules with different reactive groups (in a 1 : 1 mixture).The mean conversion factor now is P divided by the crosslinking probability of statistically distributed reactive groups, i.e. FIG. 16.- 2b i o 60 80 160 polymer concentration / w t O/O -Experimental conversion data on cis-1 ,Cpolybutadiene (0) and by Vollmert 2o explained by the meander model (theoretical polybutylacrylate (0) curves). 1 by .\/c. In fig. 16 two theoretical curves according to the formula 1 2d 1 conversion = [G - 3 - have been fitted to Vollmert’s results. The fit parameters used are q = 3,3, r/d = 5 for cis-l,4polybutadiene and q = 2,8, r/d = 3 for polybutylacrylate, which are reason- able values. For high polybutadiene content the measurements indicate an increase of the bundle diameter r/d from 5 to 8 which corresponds to the value determined by our analysis (table 3).The observable deviation of the measurements from the theoretical curves at low polymer content is certainly due to a premonitory mutual penetration of different molecules and probably defines the concentration region in which the coil model will be responsible. Support by the Deutsche Forschungsgemeinschaft and by the Fonds der Chemie is gratefully acknowledged. We thank our colleagues for helpful discussions and Mrs. Schiffner for preparing the manuscript.76 MEANDER MODEL TABLE 3.-vALUES OF VARIOUS PARAMETERS FOR DIFFERENT POLYMERS polymer (amorphous) n c c 8 3 h .- W h a 3 e x h a 8 B 3 - ~ ~ ~~~ ~mOllOmer/g mol-' 254 104 62 192 86 86 56 68 54 28 74 pjyDh./g cm-3 1.26 1.05 1.39 1.34 1.19 1.22 0.92 0.91 0.94 0.86 0.97 LmOnom,rl+ 10.5 2.22 2.2 10.8 2.1 2.1 2.3 4.05 4.3 2.4 2.8 meanchaindistance, d/A 5.7 8.7 5.8 4.7 7.6 7.5 6.7 5.5 4.7 4.8 6.7 bl Agflkcal mol-l r/d 8 t (r + x ) / A a% 8 7 180 6.4 5 75 .-.2 TglK 412 363 351 343 302 278 200 203 167 150 2 (r + x ) / A 170 150 290 140 150 140 135 135 115 150 3 ch/kcalmol-' 1.60 1.22 1.51 1.32 1.03 0.96 0.63 0.72 0.60 0.55 x (f= 1 0 - 3 ~ 4 g g, Qy/kcalmol-' 7.5 7.5 11.3 7.5 10 6.2 9.7 7.0 6.3 3 es vol10'3H~ 4 4 4 4 4 4 5 0 2 1 0 4 ~~ 5.2 4.1 3.3 3.3 2.4 1.4 0.50 4.1 43 3 335 373 298 333 353 433 298 5' 4 ' 4 f 3.5d 2 b 1.2' 0.6b 3.5g ?, c, AH:hcor/cal g-l 0.63 1.0 0.73 0.69 0.91 0.90 0.84 1.29 1.36 0.50 9.2 AH:XPlcal g-l O.gh 0.8 ' 0.8 h*k B 5 monomers per 1 2 2 1 2 2 2 1 1 2 2 < segment a - k See ref.(15). 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