# Entanglement of Thread Molecules and Its Influence on the Properties of Polymers.

код для вставкиСкачатьEntanglement of Thread Molecules and Its Influence on the Properties of Polymers By Martin HoffmannL"] Dedicated t o Professor Otto Bayer on his 75th birthday This article shows the extent to which the entanglement of threadlike molecules can influence many of the properties of deformable polymers and of solutions of macromolecular substances. Thus, the molecular concept of entanglement leads to quantitative predictions of the dependence of the entanglement number on the nature of the polymer, its concentration, and extension. Experimental values of the relaxation modulus confirm these predictions for melts and solutions. The influence of entanglement on relaxation and flow processes leads to very simple relationships for the magnitudes of the structural viscosity and the shear stress at which the structural viscosity starts to become measurable. Osmotic pressures, light scattering intensities, and diffusion coefficients of concentrated solutions can be derived quantitatively from the conformational constraints due to entanglement. Entanglement is effective above a concentration at which the three-dimensional coils fill the solution volume, and probably also at lower concentrations. At very high concentrations entangled structures may be formed which drastically reduce the number of possible conformations. The swelling and extension behavior of cross-linked deformable polymers can be understood only if entanglement is taken into account. 1. Introduction In dilute solutions['] and in melts[21long threadlike molecules demonstrably assume a conformation resembling a threedimensional coil: for a polystyrene sample of molecular weight M = 1 x lo6 in toluene solution the diameter of the coil is about 100nm. Such coils are approximately spherical and in this case have a volume V of about 3 x lo8ml per mole. The polymer concentration in the coil is M/Vand here amounts to about 0.3 wt-%. In solutions containing more than 0.3 wt-% of such a polymer the coils may interpenetrate or escape interpenetration by shrinkage. The coils are energetically preferred, i. e. stable, conformations; they rapidly return to their original shape after deformation (rubber-elasti~ity[~]). As a result, they shrink only little on concentrating the solutions, and they are so large in melts that, with such threadlike molecules and because of the high polymer concentration in the melt, the number of molecules contained in the volume of one coil must be of the order of lo2.If several coils interpenetrate with retention of the coil conformation, then chain fragments must touch one another during the continual movement of the chains (conformational changes); in some cases the movement is such that one chain twines round the other so that further movement is hindered. Such contact between two chains acts briefly as a chain junction made by main valences. This concept of physical cross-linking by entanglement was first formulated several decades ago (for a review see C41), but was accepted only slowly as a basis for theoretical interpretation of the properties of polymers and their solutions[4- 61. Such conformational interaction of polymer molecules does not arise from any specific interaction by secondary valence forces between them and therefore occurs generally in solu- tions and melts, as well as in partly crystalline and crosslinked polymers. Since entanglement influences many of the properties of high-molecular substances and their solutions, the underlying principles and effects will now be surveyed and formulated quantitatively for noncrystalline systems. 2. The Molecular Model The shape of long molecules essentially resembles a random flight path, being formed by joining rigid chain fragments of given length (segments) to one another in such a way that the direction of the last segment is completely independent of the direction of the previous one and all directions occur with equal probability[']. These segments of length / replace equally long chain fragments regarded as rigid (straight or possibly curved). The distance h between the ends of a sufficiently long chain of Z segments is then on average a vector of length $6. If Z is large, the chain can be divided into subchains each with Zimembers, and each subchain obeys chain statistics ( i . e. hi= The vectors hi at the junctions are on average perpendicular to one another, as is easily appreciated by considering the case of Z=2Zi; here h = and hj=11/2/2, so that h=1/2.hi is the diagonal of a square of side length hi. For the description below, which is correct to an order of magnitude, the segment model can be simplified to the string-of-beads model; thus, we assume that the segments are compact balls that in a melt adopt a close packing and become deformed so as to eliminate empty spaces. The segment molecular weight M s can then be employed to calculate how many segments are contained in 1 cm3 of the melt of density p ( N A z 6 x lo2')); fi). fi [*] Dr. M. Hoffmann Bayer AG, Zentrale Forschung, Wissenschaftliches Hauptlabor 5090 Leverkusen (Germany) Angew. Chem. Int. Ed. Engl. 16.751-766 ( l Y 7 7 ) 751 O n estimating how many bonds are required for the direction of a polymethylene chain to be continued with equal probability in eaEh spatial direction (a requirement of the segment model), one obtains a segment molecular weight M s of around 170, in agreement with estimates from measured coil dimensionsr8'; thus for polyethylene Ns is approximately 3 x 1021. Next, we can calculate how many contacts between different chains occur in 1 cm3 of this melt. Figure 1 shows two macromolecules A and B in a melt, the two chains being in contact at segment a of chain A. The segment a has twelve nearest neighbor sites, of which two are occupied by neighboring portions of chain A. In addition, back-coiling of chain A causes further p nearest neighbor sites of segment a to be occupied by chain A. 2 Fig. 1. Two thread molecules A and B whose spherical segments in the melt adopt close packing (represented two-dimensionally). When estimating the back-coiling number p , it must be remembered that the third segment (counting from a) can occupy, out of the twelve possible places, those four that are adjacent to a, so that 4/12=0.33 neighboring sites of a are occupied. This treatment can be continued for further segments, considering then the positions of the third segment. The residence probability P of a more distant segment at a site adjacent to a can be calculated by the known statistical function["J for chain-end separations (ais F b r y ' s expansion factor["]): (3) For polyethylene therefore, in conjunction with eq. (I), N K = 5 x 102'. It still remains to be shown which of these contacts should be regarded as entanglements. Entanglements should be contacts that withstand many changes of chain conformation and thus are so long-lived that they act as cross-linking sites, permitting rubber-elastic deformations of the melts. Figure 2 shows such a contact position of chains A and B. In this presentation['] the chains are symbolized only by the vectors hi of the subchains, which here have the designations A l , Fig. 2. Two thread molecules A and B touch at one point. Their subchains are symbolized by the vectors A , and E l .The vectors hA and he join the end points of two chain fragments that are themselves situated at further entanglement points 191. = For I= 1 each of the twelve neighboring sites has the volume d x d y d z = 1 and in eq. (2) the exponent -3/(2Za2). The summation over all values of 12P(Zi), leading to p, should already be stopped, however, when Zi is still small: entanglements are continually being untied and new ones formed; the network chains of the entanglement network, i.e. the portions of the chain between entanglement points, contain in the melt relatively few segments [see eq. (17)], of which certainly only a small proportion can be attached at segment a without severely straining the residual part of the network chain. Contact of chain A at segment a with distant segments of the same chain should therefore be regarded as contact between different chains. In solutions the network chains comprise many more segments than in melts, so that p can then be greater than in melts; p z 1 has been estimated for the latter. 752 + 3.5 NK=-N~ A P(x,y.z)dxdydz If one considers that chain A coils back to segment a from two directions, one finds that, of the twelve neighboring sites of segment a in the melt, 2 + 2 p x 4 sites are occupied by segments of the neighboring portions of chain A. The remaining 12-4=8 sites adjacent to segment a can now be occupied by segments of other chains. If chain B occupies such a position with segment b, then a and b have four nearest neighbors in common, of which on average 4 x 4/12 are occupied by chain A and the same number by chain B. Chain B thus occupies 1 (4/3) = 2.3 of the sites adjacent to a. The eight sites at a which are free for contact with other chains can thus be occupied by 8/2.3=3.5 foreign chains. On counting all NK contacts of the various chains in 1 cm3 ofthe melt, each contact refers to two segments and is therefore counted twice, giving the contact number NK asr9]: Azr B1, and B2. In the outer part of Figure 2, these vectors end in further entanglements of the chains A and B. Had the contact of chains A and B taken the form shown in Figure I , it would have been rapidly broken by the usual conformational changes. On average, such changes leave unaltered the length of the vectors A l , Az, B1, BZ and angle CI (on average a right angle) between Al and A, and the angle p between B1 and B2. Having fixed the lengths of the vectors A, and Bi,as well as a and p, the allowed conformational changes thus correspond to rotating chain A (or B) around the vector hA (or h g ) . It is now simple to formulate the conditions under which contact between chains A and B is not destroyed by such rotations. In so doing, it must be borne in mind that h A and h g can assume any position in space. The contact is not destroyed if the following geometric conditions are fulfilled (for a =/.'=90"): a) Chain B touches chain A in the internal angle of the vectors A , and A2. This happens with a probability W,= 114 (=90"/360"). Angrw. Chem. I n t . Ed. Engl. 16.751-766 ( 1 9 7 7 ) b) Vector B1 lies above and vector B2 below the plane of the vector A l and A 2 : Wb= 1/2. c) In cases where the plane containing A, is at right angles to that of B, : B points in directions in space that correspond to the octant lying diagonally "opposite" the internal angle between A l and A 2 : Wc=1/4. W, is not changed when the plane of A, is not at a right angle to that of B,. The fraction W= W, x Wb x W,=O.O31 of all NKcontacts is therefore long-lived. Such contacts are entanglements that are only destroyed when neighboring contacts are also released in a cooperative process. This argument can be extended to all cases in which the angles x and/or are not 90". If/1=90", then W2 is always 1/2 and Wi =z/360, and further W,=(180-~()/360. The probability W,, then amounts to: 8 [M( 180- .)./I(180 3604 W/j = ---- -/I)] (4) /3 can assume any values between 0 and 180°, so that the probabilities are proportional to sincc and sinj?, respectively, the maximum of the frequency distribution of statistical conformations lies at 90°, and half of all cases have angles between 60 and 120". If Wziiis multiplied by the corresponding values of the frequencies of x and and averaged, the result is WR=2/n4=0.021. Furthermore, W, is not changed when the vectors A], A2, B 1 , B2 are allowed to have different lengths. If, however, the entanglements are sufficiently long-lived, then Ai and B, "forget" to which chain they belong and thus on average point to the corners of a tetrahedron. The frequency distribution of the angles then has a maximum at 109.5", so that the probability wfalls to ~10,,5=0.018[91. The entanglement number N , is then: to this end faster than a more distant oner9].Such considerations will be taken up in Section 4 below. We would add here that, even in the absence of any long-term relaxation, the entanglement number N , given by eq. (5) is lowered if the entanglement network is subjected to shear or deformation['2]. Deformation causes changes in the angles z and /3 and thus in the geometric requirements for entangle2 / / I)/- - -X z and N , =0.01a N~ (51 For polymethylene eqs. (4) and (3) give N,,=9 x The preceding derivation neglects the effect of end groups and thus assumes infinitely long molecules. If the network chains of the entanglement network have a molecular weight M, and the molecules have a molecular weight M , then networks can be only if the condition known for cross-linking polymerizations[' '1 is fulfilled for the construction of a continuous network: In the derivation of eq. (5) it was also considered unimportant that some of such entanglements are more easily broken than others. Thus, the entanglement number calculated according to eq. (5)includes entanglement for which the pair of values z and / j or the position of chain A relative to chain B only just fulfil the geometric requirements for entanglement. Even small diffusion movements of neighboring entanglement points can break such entanglements, so that they will certainly be shorter-lived than the others. Furthermore, n o consideration was given to how easily the chains could slide over one another and to how far the end of the chain is separated from the entanglement of interest. Diffusion of a chain end could break an entanglement near Anyew. Chem. Int. Ed. Engl. 16, 751-766 ( 1 9 7 7 ) . Fig. 3. As a result of stretching the spatial direct~onsand lengths of the vectors at a (tetrahedral) contact of two chains are deformed, so that the entanglement no longer has the same probabllity W7/j [see Eq. (4)] as before the stretching [12]. ment. The tetrahedral entanglement (Fig. 3) is broken if suficiently strong uniaxial stretching occurs in directions that do not lie in the part of the spherical shell drawn in the figure (or in the opposite direction), for then the stretching directions make an angle smaller than 90" with one or more of the relevant chain vectors. Thus, purely geometrically and in a very short time, a large stretch reduces the number of entanglement points per unit volume to the fraction X : (7) If one averages over all x and /j corresponding to their frequency, one obtains['21for a maximum of the frequency distribution 90" X=O.33 and for 109.5" an almost identical value. If one now permits even a slight relaxation or slip of the sometimes very unevenly stretched network chains at the entanglement points, then the number of residual entanglement points decreases almost to zero. Furthermore, it is important to know how small extensions reduce the entanglement number N,. The ends of the vectors A', A 2 , B1,and B2 can occupy any desired point on the surface of a sphere of radius IAl. On deformation (extension ratio 2 = L/Lo)such a sphere is transformed into an ellipsoid of rotation ( a = I A l . i , h=IAIfl.), so that on average each vector assumes a greater length IA,1[31: 153 The coordinate x (in the stretching direction) and y of each vectorend-point have thuschanged i n t o x ' = x . i and y ' = y f i so that an angle y formed initially by the x-axis and such a vector (tany=y/x) changes to ;*' (tany'=y'/~'=tany/i'.~). If one puts the length 1Aj.l into the equation for the ellipsoid of rotation one obtains the relevant angle x' between two vectors as tan(c('/2)= f1/3.'.'. Before the stretching, the angles a and amount on average to 90" and are transformed into smaller or greater angles N' and p'. As a result, WxP[eq. (4)] decreases to Wx.lI., i. e. the entanglement number N , falls to N:. The factorf;. (= W g o , g o , j / W ~ ~ gives , ~ ~ , the ; . ~fraction l) of entanglements that are still present at i. (relative to the number at ;"=I) (Fig. 4a). The elastic restoring force of an entanglement network (referred to the cross section of the stretched specimen) is therefore not, as in main-valence networks, proportional to 2' - 1/2 [see eq. (9)], but is instead proportional to f,.(;.* - I/>.). Figure 4 b shows the course of this function. Above a certain extension the restoring force should decrease on further extension. The entanglement network thus breaks on deformations around 1= 2. The experimentally observed extension Law for non-cross-linked, very slowly relaxing polymers confirms that the entanglement number N , decreases with 2 in the manner derived here[l21. Moreover, the shear stress at which structural viscosity occurs shows that entanglement networks break at specific extensions (see Section 5). that, although a neighbor along the chain, is not the junction nearest in space but in fact one further removed (Fig. 5) (see also Section 8). If such a network is stretched, volume elements of interest become more remote in the stretching direction. b R Fig. 5. At high concentrations the distances between the entanglement points are smaller than those between the ends of the network chains. Then several simple networks seem to interpenetrate (different point symbols for each network). If such a system is deformed, contacts are established that were not previously present; these contacts are not broken during extension, thus acting as additional entanglement points [IZ]. Since the contour length of the network chains remains unchanged, the network chain finally contacts a spatially close network junction["' (Fig. 5). Even a small extension can lead in this way to larger entanglement numbers, estimated["] as (S/3)N , , = 1 . After a small extension, therefore, the N , , = in eq. (5) should be replaced by (S/3)N,,,= 3. Conclusions from the Molecular Model and Their Experimental Testing with the Aid of the Relaxation Modulus Equation (5), the result of the molecular model, can be tested by investigating whether melts have highly elastic properties. As shown in Figure 6, it is in fact found experimentally that above the softening temperature a rapid deformation is followed by restoring forces cr that remain constant for a longer time (plateau of the relaxation modulus in the rubber region of the relaxation). Use of the extension ratio i= L/Lo permits calculation of the relaxation moduli E, or G,=3E, (G,=shear modulus in experiments withf;.z I 7 2 x- 3 (9) --.. 2 x- .- .. 3 Fig. 4. a) The percentage of active entanglements as a function of the extension ratioi.f,.= W90,so,;./W90.90,;.=, is not exactly I/;. as required by the MooneyRivlin equation [19]. b) The restoring force ur of an entanglement network divided by the relaxation modulus G , (at .; = 1) in dependence on the extension ratio i. [cf. eq. (9)]. If iis larger than corresponds to the curve maximum, an equilibrium retractive force is no longer established: the sample flows by a mechanism different to the usual one, where the conformational changes destroy and rebuild entanglements even at low T, The relationship applies for all polymers. The above considerations neglect the fact that in entanglement networks of undiluted polymers (and in very concentrated solutions) the network chains lead to a network junction IS4 At sufficiently high molecular weights G, is proportional to 2N,, the number of network chains per cm3 of the melt, and it has a finite value only when the molecular weight M , ( M , = number-average molecular weight of the polymer) of the polymer exceeds 2 M , (Fig. 6). However, at appreciably larger M , G, is independent of M . This result is to be expected from the description of entanglement given above, because at M < 2 M , the known requirement for network formation [see eq. (6)] is not fulfilled. On the other hand, if 2 M , e M , the entanglement number does not depend on the molecular weight because the small number of chain ends does not influence the relaxation modulus. For polyethylene melts at 120°C one obtains according to eq. (9) a modulus G, of about 1.1 x 106Pa, and thus an entanglement number N , = 8 x 10' at 25 "C. The agreement of this value with that estimated from eq. (4) is much too good in view of the uncertainty in the segment size, the crude approximations of the molecular model used here, and the uncertainty in the conversion from 120 to 25°C. The experiment thus confirms the molecular model at least t o within Angew. Chem. Int. Ed. Engl. 16,751-766 ( 1 9 7 7 ) 10’ an order of magnitude and encourages us in our search for further conclusions. In proceeding further it must be remembered that the segment model correctly represents the conformations of a chain of CH2 groups linked together at the usual valence angles. The auxiliary quantity “segment” is used only to simplify the statistics. In reality the spheres considered here are the CH2 groups of the polymethylene chain and these are joined together with specific valence angles. If to a polymethylene melt we add oligomers or solvent molecules having the size of one segment (or the size of the CH2 groups), not changing the chain conformation by special interactions with the polymer, then the solvent molecules will occupy some neighboring sites of each segment and thus reduce the number of contacts between different chains (contact number NK).The probability of pairs of segments from different chains occurring is known to be proportional to the square of the volume fraction cp of the polymer, provided that the back-coiling number p is not dependent on cp: *\ \ lo8 \ \ \ \ \ -t \ - lo7 2 lo6 v ae. \ \ - - -- 10’ - strong cross-linking weak cross-linking -- 10‘ M = 0.5 1.5 5 15.105 NK= , ~N K , .‘p2 I ts1Fig. 6. Restoring forces ur arise after momentary stretching of cis-1,4-polyisoprene, which decrease as the load-application time is increased. In non-crosslinked samples of molecular weight M > 2 x lo4 it is observed that G , (the relaxation modulus calculated from the restoring force) remains nearly constant for a certain time (the plateau of the modulus in the rubber elastic region of the relaxation). G, is there almost independent of the molecular weight and of weak cross-linking by main valences [14]. (10) t Since, furthermore, the geometric conditions of chain conformation at entanglement points are not changed by the presence of solvent molecules, it follows from eq. (10) and (5) that the entanglement number is[’]: N,,,= Nv31 .‘pz (1 1) Table 1. Relaxation moduli G , , critical shear stresses T* (see text), molecular weights for network chains M,, and molar volumes V, of the repeating unit for the ciitanglenient network:, i n linear polymers, in dependence o n thc volume fraction q and temperature. Polymei Polymethylene Polybutadiene (Li) Polyisoprene (cis-1,4) Polyisobutylene Polydimethylsiloxane Poly(viny1 acetate) Polystyrene Polyibutyl acrylate) Poly(2-ethylhexyl acrylate) Poly(dodecy1 methacrylate) Polyisoprene (Li) (M=1.7~ 10’) in toluene Polystyrene ( M . = 6 x 106. c‘, : 0.3) iii toluene in decalin Polystyrene ( M = I . l 5 x 10‘) in toluene in decalin Angew. Chum. Int. Ed. Engl. 16,751-766 ( 1 9 7 7 ) + 25 i 25 + 25 + 25 f 25 +25 +25 i 25 i 25 25 -51 - 22 i25 84 25 25 i 25 + 25 125 25 i25 25 +25 25 25 25 25 25 25 i 25 i 25 i 25 i 25 + + + + + + + + + + + + + 25 + 25 31 32 37 62 72 73 91 129 203 285 37 91 97 91 1 1 1 1 1 1 1 1 I 1 1 1 1 1 0.59 0.44 0.32 0.26 0.20 0.19 0.17 0.11 0.055 0.21 0 13 0.12 0.059 0.028 0.12 0.058 0.028 0.25 0.12 0.2s 0.12 90 -.. 55 : 70 32 20 10 10 5.5 4.0 3.0 1.7 28 29 30 29 10.5 1.9 3.5 1.45 0.70 0.90 0.55 0.13 0.012 0.56 0.28 0.24 0.055 0.006 0.16 0.035 0.005 0.76 0.06 : 30 0 75 0.050 5 30 11.0 8.0 9.0 3.0 3.0 I .I 2 500 4 300 7 300 I I 000 25000 30000 45 000 65 000 ROOOO 140000 17 5.3 I .5 0.4 0.15 0.01 5 0.40 0.15 0.1 5 0.045 0.008 0.14 0.03 0.70 0.14 0.60 0.14 155 Table 1 shows that the relaxation modulus G,, which according to eq. (9)is a measure of the entanglement number N,, does in fact depend on the square of the volume fraction cp of the high-molecular components in solutions and polymer mixtures[’]. The same result can be derived from the right-hand side of Figure 7 K-G,). Fractions with molecular weights M<M,,,=, thus lower the entanglement density of a polymer. 1.0 1 0.5 1 - 10.1m -2 0.05 - 0 O.O1I 0.005 . 15 % 5 v, 0.001 t lo-’ 100 10’ - lo2 t [SI 10’ Fig. 7. Restoring forces arising after momentary (5 ms) deformation of mixtures oftwo molecularly homogeneous (Uy<0.02) polyisoprenes (Li) ( M I= 2 x 10‘. M 2 = 1 . 4 x to5) plotted as a function of the load-application time at 22°C. f;fa has the following meaning: f=u/;, is the restoring force referred t o the initial cross-section of the sample, and j . is the same for all samples (3.0bar=Cr; eq. (9)). According to eq. (9), ur-Cp The components relax at different rates, which leads to the stepwise nature of the relaxation curves of such mixtures [9]. The numbers on the curves are the proportions of M I in the mixtures. When we next consider melts of chains whose side groups are so large that they occupy neighboring sites of a segment, and thus decrease the entanglement number N,, we see that, as in eq. (II), the square of the volume fraction of backbone chains determines the entanglement number. If V,,,, is the molar volume of an ethylene unit and V, that of the repeating unit of a vinyl polymer with side groups, then we should find that[’]: Figure 8 and Table 1 show that eq. (12) is also confirmed by experiment[’]. In these calculations we used the molar volume V p of the repeating unit of the polymers or the molar volume of a chain fragment of constant length, this fragment corresponding to a repeating unit in the case of vinyl polymers. In eq. (12) it is assumed that the backcoiling number p does not depend on the structure of the polymer; special hindrance to rotation is also disregarded. In place of the number N , of entanglement points in 1 cm3 we may use M,, the molecular weight of the network chains in the entanglement network (p=density of the polymer). From equations (9), (1 I), and (12): - 50 100 10 [rnl.rnol-’] 500 Fig. 8. Relaxation moduli C,, measured according to Fig. 6, of various polymers (for individual values see Table 1) plotted as a function of the molar volume of a chain fragment whose length corresponds t o a repeating unit of the vinyl polymers [9]. A proportionality between M c , , and l/cp was earlier derived by Bueche by a different route and was tested e~perimentally[~]. At this point we should turn to yet another consequence of the molecular model, concerning the segment size. Eq. (3) shows the relation between the segment number N s and the contact number NKin chains without side groups. According to eq. (12) side groups reduce the contact number proportionally to (K,pE/VE)2.It is now possible to define an apparent segment size by assuming spherical segments (larger than in polyethylene because the spheres now contain the side groups) and apply eq. (3) for the calculation of the segment number. The segment number in melts is then given by: It follows that the molecular weight of the segment Ms= & p in melts is: Experimentally, it has been found that eq. (16) is confirmed, although the factor 0.36 is better replaced by 0.2[91. Now, since according to eq. (13) the molecular weight Mc,t of the network chain in the melt is also proportional to V:, the network chains of different polymers in the melt all contain the same number of segments[’l: M ~ =,-~ ~ 4.1 __ = 11 to 21 MS,, 0.36 to 0.2 For nonideal solutions eq. (14) must be modified when the distance h between the ends of a molecule in such solutions is greater by the Flory expansion factor[131Y. than in ideal 756 Angrw. Chrm. I n t . Ed. Engl. 16. 751-766 ( 1 9 7 7 1 solutions (denoted by a subscript 0): h=he.s(. The volume occupied by a chain fragment of molecular weight M , in the form of a coil is then increased by the factor ctf over that in ideal solvents, so that correspondingly more foreign chains pass through this coil and form entanglements. Moreover, in each volume element there are s(," as many chains, because at a given total concentration the more coils must overlap, i. e. interpenetrate, the greater the coils are at a given molecular weight, so that the total concentration may everywhere be attained. Finally, the segment length also increases, under the usual assumptions ( Z o16aZ= Z 'I and Z olo = Z 1) by the factor a:. The segment molecular weight M s thus similarly increases by about 12, so that N K and hence N , should decrease by zf. In all, z, thus changes eq. (14) to['41: The experimental relaxation moduli confirm (see Table 1 and Figs. 9 and 19) the influence of the efficiency of the solvent and thus the influence of x,. aoot 0.01 9- 0s 1.0 Fig. 9. Relaxation moduli Gr.wdivided by the concentration (=volume fractionjplotted as afunction of theconcentration for mixtures of polyisoprene (Cariflex IR 305, M , , = 1 . 7 x lob, U,<O.I) and toluene, as measured from the relaxation behavior IS -3000ms after momentary (5 ms) compression o r shear and rxtrapolatcd to t = O (Hottinger technique, display oscillographj. The osmotic presrures pox, also plotted were measured with an automatically compensating osmomeler [I41 (left-hand branch of the curves) or taken from the literature [29]. According to experimental investigations of structural viscosity["] and relaxation modulus (see Table 1) in a temperature interval of about I O O T , the temperature dependence of Mc,l is negligibly small. The modulus plateau can therefore hardly Angew. C h m lnr. Ed. Enyl. 16, 751 766 ( l Y 7 7 ) be due to an equilibrium between association and dissociation of the polymer 4. Influences of Entanglement on Relaxation and Flow Processes All the investigations described so far show that the entanglement number is an intrinsic property of each polymer and is thus independent of time. Since, however, according to Figure 7 restoring forces disappear in the course of time and melts or solutions can flow, it must be assumed that the entanglement numbers measured correspond to a dynamic equilibrium between disentanglement and renewed entanglement. The underlying processes are not specific associations and dissociations of the polymer chains but the usual conformational changes of the chains, diffusion of chain ends through entanglement loops[6], and cooperative and nearly simultaneous conformational changes at several neighboring entanglement p o i n t ~ [ ~ -It~is] .found that polystyrene powder ( M w zlo'), produced by spraying of dilute solutions into vacuum, has the usual flow behavior immediately after pressing in vacuum, i.e. about 40min after the spraying. Thus. the entanglement network must have formed in the course of this time. Other experiments show that large dissolved molecules diffuse away from one another in milliseconds after instantaneous fission; then the fragments display the usual light scattering" 'I. The conformations clearly change so fast that even complicated structures such as entanglement networks are formed sufficiently rapidly. Further information on this is provided by a study of strain relaxation and of flow. Figure 7 shows relaxation measurements made on mixtures of polyisoprenes with a heterogeneity index of about 1.1. It is evident that these mixtures form two kinds of entanglement networks. At short times one finds moduli that correspond to the usual moduli for polyisoprenes (Fig. 6), independently of the composition of the mixture or its molecular weight M , (weight average molecular weight). This means that the two types of molecules have formed a common network whose network density is independent of the composition of the mixture. At long times, however, there is a second plateau of the relaxation moduli that shows much smaller moduli and depends strongly on the concentration of the high-molecular fraction and can therefore be assigned only to entanglement of the larger molecules. At this time all the entanglement positions in which short molecules take part have changed their conformation, their contribution to the restoring force being lost because the chains have formed an equal number of new entanglements with the use of undeformed chain fragments. The entanglements of the larger molecules remain; these are broken only after longer times, because there must be a cooperative process involving many entanglement points simultaneously-or at least coordinated in time and spaceand this cooperation requires the more points, the larger is the molecule concerned. Buechd'] and others (for a review see l41) have used the concept of entanglement in theories of flow. Accordingly, the maximal relaxation time t , in the rubber-elasticity region of the relaxation and the viscosity q proportional to it depend extensively on the molecular weight M and the concentration cp (volume fraction)" '1: t , =const.tT =const. kf3.'.(oJ (20) 751 Figure 7 gives the times in which the entanglement networks of molecules of various sizes undergo strain relaxation to one-tenth of its original value; these times are approximately in the ratio (M2/M1)3.5, thus confirming eq. (20). Furthermore, the step heights obey approximately the expected law, namely eq. (13). On the other hand, concentration dependence of t , in this experiment appears to be considerably weaker than predicted by eq. (20); the reason for this is that the process cannot be described by a single relaxation time, and the maximal relaxation times t , for disappearance of the last traces of a restoring force cannot yet be measured with accuracy. However, many investigations of the flow of solutions confirm the concentration dependence expressed in eq. (20) (Figs. 10 and 11). Naturally, this is only so if the prerequisites for eq. (20) are satisfied, i.e. if the expansion factor x and the back-coiling number p as well as the segment mobility are none of them strongly dependent on ( P [ ~ . ' ~ I . The flow behavior of melts and solutions containing entangled molecules also shows other effects of entanglement. It is found that the molecular volume V, of the repeating unit of the vinyl polymers has an appreciable influence[' 71: The factor f ( T- T,) describes the influence of the strongly temperature-dependent segment mobility. In the absence of entanglement the viscosity is far less influenced by M , cp, and &["I: I . . . 10 1 c [g /lo0 rnll- 01 Fig. 10. Relative viscosity q/qtOl (from flow curves extrapolated to shear stress T =0) plotted as a function of the concentration for polystyrene fractions of various molecular weights. [q] =limiting viscosity in [dl/g] in toluene at 25°C. Above a critical concentration the viscosity of the solutions increases with c4. The critical concentration decreases with increasing molecular weight. Thus, eq. (21) holds for the dependence of the viscosity on molecular weight at molecular weights M that fulfil the network-formation condition [eq. (6)], whereas eq. (22) applies if M is smaller. This kink or transition in the logq-logM relationship has been shown to appear in many polymers and solution^[^^ ', 7 32 1 1 . M., 4.3.106 i 10: 0 10' 6 F 'I g, -+/-+ P' -1 10. 0 a m F 10' /' 4 1I =5570 ' 10' 2 10' (ED2 MI3 M)5 01 Q2 0.3 45 cpFig. I I . Flow curves for mixtures of two polyisoprenes (Li, M 1 = 1 . 4 x lo5 or 1.2 x 10'; M 2 = 2 x [ O h , UM<O.l)give viscosities q i = o and q T = x extrapolated t o shear stresses r=O and T = J ,respectively, which are here plotted as a function of the concentration (=volume fraction m) [17]. The h = z . is extrapolated as if the viscosity q 1 of the lower-molecular component were independent Of T and at sufficiently small q , is proportion t o 'p2.At Sufficiently small q , , however, ~ i is ~ proportional = ~ to q4. q l = l . l x lo5 refers to M 1 = 1 . 4 x 1 0 5 ; ~ 1 1 = 5 . 510' x refers t o M I = l . 2 x 1 0 4 . 758 103 lo4 T [Pal+ Fig. 12. Flow curves of unbranched polydodecyl methacrylates at 25°C. The flow curves of these polymers can be measured down to the vicinity of the second Newtonian region without fracture of the melt [17]. Anyrw. Chem. l n t . Ed. Engl. 16,751-766 ( 1 9 7 7 ) A further consequence of entanglement is the appearance of a structural (non-Newtonian) viscosity that is much stronger than in systems where the thread molecules d o not entangle. If melts or solutions that follow eq. (21) are made to flow under higher shear forces, their viscosity decreases with increasing shear stress 7 (Fig. 12). At very high shear stresses the viscosity reaches values that correspond to those of eq. (22)[17],so long as the melt does not previously collapse (melt fracture). The quotient of the viscosities qv and q constitutes a suitable measure of the magnitude of the structural viscosity: Figure 13 shows flow curves for polyisoprene mixtures[g1at low shear stresses. Measurements on solutions["] and on the mixtures of Figure 13 are shown in Figure 14 to conform, at least approximately, to eq. (23) which, as previously, is formulated for constant expansion factors a, and constant back-coiling numbers p . The influence of the basic molar volume V, is not clearly visible for the few polymers studied, lo5- I I 10' J 0I 0.01 1.0 v Fig. 14. q r = ~ / q r = r as a measure of the magnitude of the structural viscosity plotted against the concentration for polystyrenes ( M = 1.4 x lo', Uhc10.3) in toluene (o), and for polyisoprene ( M = 1.7 x lo", U M s O . l )in polyisoprenes of lower molecular weight ( 0 ; see Fig. 13) as well as in toluene ( + ) at 25°C. The concentration dependence sufficiently confirms eq. (23). To obtain a visible structural viscosity of the melts it is necessary to use various shear stresses, suited to the individual polymer. For a more precise discussion we use the shear stress z, found at the inflection point of the plots of logq uersus logt. Figure 12 shows that the z, are almost independent ofM, so that zwmay be regarded as a property of entanglement networks. Furthermore, the r, follow laws similar to those for Grr9, 'I. Eq. (24) summarizes the results for several polymers at various concentrations (see also Table 1): r 10' I lo4 t [Pal- lo5 Fig. 13. Flow curves of mixtures of homogeneous ( U ~ c 0 . 1 polyisoprenes ) ( M , =1.6x 10'; M 2 = 1 . 4 x lo5) at 25°C. Addition of the component with a high molecular weight raises the viscosity qr=o and produces a structural viscosity and an inflection point in the flow curves in the region of T < lo5 Pa, this inflection not being observed with homogeneous polymers [9]. The numbers on the curves give the proportion (percentage) of M , . apparently because other factors such as differences in Bueche's slipping factor['] may play a part. The constants derived experimentally[17Jfor eq. (23) agree with the constants obtained differently for eq. (14), in fact much too well in view of the large limits of error in the construction of eq. (23). The size q T = o / q T = xof the structural viscosity is thus simply equal to the square of the number of network chains per molecule, which is a clear indication of a cooperative process. The experimental result does not confirm other theoretical claims so we11[4. 51. Angew. Chem. Int. Ed. Engl. 16, 751-766 ( l Y 7 7 ) This finding can be interpreted as showing that the same distortion of the entanglement network is always sufficient to reduce its effect on viscosity by a certain fraction, confirming the considerations of the dependence of the number of active entanglements on the extension, which lead to the assumption that rWz1.6G,(see Fig. 4). The collapse of the entanglement network becomes evident when, at high shear stresses, melt fracture occurs. Consequently, these critical shear stresses are proportional t o cp2/V;[l7 , 8 J so as no stretch crystallization occurs. Moreover, it can be assumed that under the influence of an applied load, i. e. deformation of the chain fragments taking part in entanglement, the diffusion steps or conformational changes leading to a removal of the load are preferred to others and so accelerate the disentanglement of the stressed network chains[6,41,but leave the process of new entanglement unaffected. A gradual decrease of entanglement could therefore occur during relaxation after large deformations. ' 159 99 98 95 90 - 10 180- T 0 ??So- -. - 8 N 5 c =4020 - 10 - 5ZL I -2 -1 I I 10 1 log 2 3 L t [sl- Fig. 15. Rate of strain relaxation (as a percentage) in a non-crosslinked polyisoprene ( M / M o " 130) after momentary (5 ms) compression increases with increasing deformation as a result of the nonlinear form of eq. (9) and for the same reason a s causes structural viscosity (see text). i.=extension ratio. f, = uJ;. can be calculated for i 2 0.92 by means of eq. (9) with Gr,== 3.0 bar. cm *. The same causes as lead to the appearance of structural viscosity lead to the fact that the relative rate of stress relaxation is greater after large than after small elongations: According to Figure 15, the halflife of mechanical relaxation decreases with increasing 5. Influence of Entanglement on the Extension of CrossLinked Materials The fact that the effective entanglement number depends on the extension is also important for the properties of materials such as rubber"'] that are cross-linked by main valences. Figure 6 shows their relaxation behavior in comparison with that of non-cross-linked material. With slightly to medium cross-linked samples we find, at short loading times, relaxation moduli that correspond to those of non-cross-linked polymers. Only after prolonged loading times does cross-linking by main valences become noticeable, so that analytic stretching experiments are often carried out with extended loading times. On the other hand, in chemically weakly cross-linked samples and with brief and slight (e. g. periodic) deformations, restoring forces are found that arise only from the entanglement network and, according to Figure 6, are usually greater than expected on the basis of the weak main-valence cross-linking. Since, however, according to Fig. 4, entanglements are made ineffective by strong deformations, even with short-lasting load applications one can, by using strong deformations, recognize the restoring forces that arise from the main-valence cross-linking. At large extensions one then finds the extension behavior predicted by the theory for main-valence networks" '1. With medium extension rates and deformations one must divide the restoring forces by the Mooney-Rivlin-Saunders procedure["] into 2 C1, the fraction due to the main-valence network, and 2 C2, the fraction due to the entanglements (Fig. 16). 760 I 0 I I I I I I , 0.5 111- Fig. 16. As described by Mooney-Rivlin-Saunders [19], the quotients of the stress u (referred to the cross-section of the stretched sample) and i z-(l/;.) (;.=extensionratio)are plotted against 1,;'. The intercept 2 C , on the left-hand ordinate is a measure of the main-valence cross-linking; that on the right, 2Cz, is a measure of the concentration of entanglements active under the measurement conditions. 2 Cz decreases with increasing duration of stress 1121. Curves 1 to 6: d i / d t = 1 3 , 1.25, 0.12, 0.025, 6 . 2 5 ~ 1.25 x The ordinate intercept of the straight part of the curve at i= 1 is 2C2, and that at i = 03 is 2 C1. The fraction 2C2 decreases, as expected, when the loading times are increased, and also if the sample is swollen, as would be expected from the strong concentration dependence of the entanglement number shown by eq. (1 3). The experiments confirm the following In the rubber region of relaxation behavior the breaking stresses and maximal extensions also rise with increasing rate of extension, because with short loading times (rapid stretching) the restoring forces affect more contacts and entanglements than with long times. Finally, the maximum degree of swelling of cross-linked samples is influenced by entanglement of the thread molecules because entanglement changes the thermodynamic properties of the solutions. In the usual swelling equations a value of the interaction parameter (Flory-Huggins parameter) i[l should then be which is influenced by the effects of entanglement, as will now be shown. 6. Significance of Entanglement for the Thermodynamic Properties of Solutions If melts or solutions become rubber-elastic by the entanglement of thread-like molecules and thus possess an entropydetermined memory, and if the entanglement density depends Angrw. Chem. I n t . Ed. Engl. 16,751-766 ( 1 9 7 7 ) on the concentration, then on dilution of such a network an entropy change AS occurs that differs from the entropy change for unentangled dissolved polymers[2o,14]. This can have a considerable bearing on the thermodynamic properties of such systems and will be formulated here, because previous formulations[2' 1 of statistical thermodynamics cannot yet be used for experimental testing. If a chain comprising b bonds or Z segments (of bonding number b, and length I = 1 ) is fastened at one end in such a way that the orientation of the first segment with respect to the anchoring is defined, then that chain can assume wo conformations, which can be calculated when the rotation potential has three minima and thus each bond can be oriented in a total of three spatial S, = k N z Inw,,, =k N 2 { b,Zln3 +(m,- l)[ln0.074- lSln(Z/rn,)- l n ~ ~ , , ~ (31) ]] At least in the case where cl,,,=l, eq. (31) agrees with other formulations for network-formation In addition to the entropy of mixing S,-S1 derived from eq.(31),afurther term - k N l l n ( l --)must beintroducedinto the equation for the free energy of mixing (subscript M), since this describes the amount of entropy that should be assigned to a giant molecule containing N 2 . Z segments (the entanglement network)[22]: A G M = A H M - T A S M = k Ti N19- T [ S , - S , - k N , In(l-cp)] (32) The osmotic pressure pOs can now be calculated by substituting eq. (31) into eq. (32) and by partial differentiationlZ2]: If such a chain is fastened at both ends, e.g. at entanglement points, then only a fraction of the previously permissible conformations is possible. This can easily be understood by considering a chain of two segments. The end points of the second segment mark out evenly the surface of a sphere of radius 1. On average, the first and the second segment make an angle of 90" to each other. If we specify that this angle is strictly retained, so that the chain ends always maintain the statistical mean value of their separation, then the end of the second segment can occupy only positions on an arc and no longer on the whole surface of the sphere. If the separation h of the ends of the chain considered in eq. (26) has the statistical mean value, then h;=Z1'; and with / = I and eq. (2) the conformation number we of the chain fixed at both ends[31is given by: (33) Vl is the molar volume of the solvent, N ~ = 6 x and k is Boltzmann's constant. To carry out the differentiation we need dsc:,,/dp [eq. (30)], ~ 2 [eq. , (19)), ~ and finally the quantity (a(PIaN1)N2: (34) We now obtain a relationship for pOsthat involves not only the known term containing the Flory-Huggins parameter z1[22], but also a further term that depends on the relaxation modulus Gr,, of the solution, i. e. on its entanglement number N, [see eq. ( 9 ) ] . f ( q ) is the experimentally determinable function defined in eq. (18). In the case of greater entanglement each polymer molecule with its 2 segments is divided into m + 1 = ( M / M , ) + 1 parts, of which m - 1 parts are network chains with Z / m segments and two parts are end fragments with Z/2m segments. The total number of conformations of such a chain is Q,: (35) For calculating the dependence of wm on concentration we require the concentration dependences of rn and x,. According to eq. (18) we have: The concentration dependence of d,, follows according to eq. (19) from that of G,,,, cp, and f ( c p ) : The entropy S , of the network of N 2 polymer molecules and N, solvent molecules can now be obtained['41: Angew. Chum. I n t . Ed. Engl. 16, 751-766 ( I Y 7 7 ) In the last term the part beginning with 1/(1 +...) describes the usually small influence of the unentangled end fragments of the network. Eq. (35) is valid only for the case of an entanglement network and must not be extrapolated to q =0, i. e. to a very dilute solution. With the experimental value of Gr,q we can calculate, by means of eq. (35) the portion of the osmotic pressure that corresponds to the term with Gr,,,, This pressure differs little at qn<O.l from the experimentally measured pressure (see Tables 2 and 3), so that we must conclude that at concentrations below 10% the first term on the right in eq. (35) is appreciably smaller than the term with Gr,cp.The parameter must therefore have values of around 0.50, e . 8 . 0.51 for polystyrene in toluene and 0.57 for polystyrene in decalin. 761 Table 2. O s m o t ~ cpressures of concentrated solutions at 2 5 T , given as rise heights d o f t h e solvent at 25°C [q]=viscosity Polymer Mw UM Solvent Polystyrene 9.0.105 <:0.02 Toluene 9.0. lo5 < 0.02 Dimethylformamide 9.0.105 < 0.02 Methyl ethyl ketone 9.0.105 Decalin < 0.02 Polybutadiene (Li) Toluene 1.5 3 6 9 18 20 30 40 25 35 55 5 10 20 40 2 4 5 10 0.07 0.20 0.65 1.25 5.4 2.55 5.18 9.35 2.8 6.8 19.9 0.19 0.47 I SO 5.30 0.58 1.25 1.73 4.93 20.57 119 0.9 1 3.51 16.31 0.21 0.56 1.07 2.45 10.22 7.75 8.0 6.9 15.8 7.94 17.55 12.28 0.70 3.40 8.58 0.13 0.44 1.68 2.35 11.6 2.74 13.45 18 40 Polyisoprene (Li) (IR 305) <0.1 Toluene 0.3 Toluene Polyisobutylene 5 10 20 4 6.7 10 I5 Polylvinyl acetate) 0.3 Toluene Polydimethylsiloxane 0.3 Toluene Poly(buty1 methacrylate) 0.3 Toluene Poly(2-etbylhexyl acrylate) Poly(dodecy1 methacrylate) 0.3 0.3 Toluene Toluene 10.1 Cyclohexane Polyisobutylene 0.3 Cyclohexane Crepe [a] 0.3 Benzene Polyisoprene (Li) 30 29.6 30 20 30 20 30 30 10 20 30 0.5 1 .o 2.0 10 20 10 20 230 119 117 98 221 750 296 101 110 74 100 320 730 580 720 [a] Polyisoprene (ci5-I ,4). Table 3. Comparison of calculated and measured osmotic pressures Polymer Polystyrene Toluene (M,=8x106) Decalin Polystyrene Toluene ( M , = l . l x 106) Decalin Polyisoprene (Li, M,= 1.7 x lo6) 762 Toluene 0.25 0.12 0.06 0.028 0.01 0.25 0.12 0.06 0.028 0.01 0.25 0.12 0.06 0.25 0.12 0.06 0.50 0.20 0.10 0.05 0.01 79 24 5.5 0.6 - 80 16 3.7 0.5 - 1.5 1.9 2.2 3.1 - 1.9 2.0 2.3 2.7 ..- 0.39 0.19 0.066 0.016 ~ 0.55 0.15 0.051 0.01 1 ~ 0.45 0.18 75 1.8 4.3 -_ 2.2 5 4.4 0.16 76 6 - - - 900 88 2.0 2.8 3.3 3.5 10 0.9 -. .- 0.61 ~~ 6.1 1.35 0.25 0.038 ~ 2.0 0.29 0.063 0.013 0.0016 1.5 0.12 0.015 0.0019 0.00012 2.0 0.29 0.063 1.5 0.12 0.015 27 2.0 0.39 0.083 0.0032 0.50 0.51 0.52 0.53 ~ ~ 0.080 0.01 7 0.0020 0.5 1 -. 0.55 0.58 0.58 -_ -. 0.50 0.51 ~ 0.51 0.56 0.0124 0.0 t 3 0.00009 - 0.054 - - 0.42 0.51 0.48 0.44 ~ 0.19 0.0062 Angel*'. Chem. In[. Ed. Engl. 16,751-766 ( 1 9 7 7 ) it follows that K*c/Ro is independent of M and proportional to c, i.e. Ro is independent of both M and e[20,141 (Fig. 18). Then the diffusion coefficient D measured in concentrated solution is also proportional to c, whereas the value obtained by measurement in the supernatant dilute solution in an overlaying experiment is almost independent of ~ ~ ’ ~Thus, 1 . the A theoretical f o r r n ~ l a t i o n ~ corresponding ’~~, to the lattice presumes that at high concentrations and with a homogeneous segment distribution the osmotic pressure is independent of the molecular weight and is proportional to the square of the concentration, but with these corrected x1 values this theory gives pressures that are much too low. For the same reason, the excluded-volume theory cannot explain the experimental values, especially as according to this theory the pressures would be expected to depend on the molecular weight. Experiment[”] shows (Fig. 17) that above a critical concentration ‘p* the concentration dependence can no longer be described by the virial coefficients of this theory. I--0.001 c [g.cm-’] Oa01 - 0.01 [g.cm-31 0.01 0.1 Fig. 18. Light scattering Ro (extrapolated to the direction of primary beam) as a function of the concentration c for polystyrenes of various molecular weights ( U M s 0 . 3 )in toluene at 25°C. At sufficiently high concentrations the scattering becomes independent of the molecular weight and of the concentration [2O]. 0.l Fig. 17. Osmotic pressure p.. (represented as the rise height d of the solvent for {)=OX?) plotted against the concentration c in toluene (polymers 1 t o 5 ) [20] and in decalin (cis-trans mixture) (polymer 5 ) at 25 “C for the following polystyrenes: I. M,=4.1 x lo4. UM<0.02; 2. M W = 1 . 7 xlo5. Uu<0.02; 3, M,=4.7x105, UM<0.02; 4, M 2 = 9 x 1 0 5 , UM<0.02; 5, M,=1.4x107, UM10.3.At sufficiently high concentrations the values reach the limiting straight line with all molecular weights. The term including Gr,q in eq. (35) is almost proportional to 40’. The same is true of xl q’, so that the two terms can be treated together by means of an effective parameter x ~ , The Xl,eff measured at high concentrations thus contain an entropy component that is derived from the entanglement network and in nonideal solutions is dependent on concentration, because there G,,$qZ is not quite constant but decreases with increasing concentration. z ~thus , increases. ~ ~ ~ Eq. (35) is also important for light scattering Ro (in the direction of the primary beam) and for diffusion in concentrated solutions, as shown by the following relationship^['^^ 2 6 ] ( K * = light scattering constant): (37) ~ ~ concentrated solution increases its volume, i. e. it swells like a cross-linked polymer. Permeation of low-molecular substances through polymers, diffusion in melts (adhesion), and sedimentation are also altered by the formation of an entanglement network, as can be recognized from the preceding discussions. Finally, the effects of entanglement on the demixing behavior of concentrated solutions, e. g. on concentration of the concentrated phase and on swelling of substances crosslinked by main valences, can also be formulated by means of the equations given above. In an approximate manner, ~ . an effective interaction parameter can be used for this purpose[14! 7. Effects of Entanglement at Medium and Low Concentrations It now remains to demonstrate at what concentrations q** a solution begins to form an entanglement network. Figures 9 and 19 show that an osmotic pressure independent of the molecular weight can be observed already at small concentrations where one cannot detect the very small rubber elastic modulus arising from an entanglement network. The limiting concentration for the formation of an entanglement network follows from equations (9) and (18): Since according to eqs. (35) and (13) pOs is independent of M and is proportional to the square of the concentration, Angew. Chem. Inr. Ed. Engl. 16, 751-766 ( 1 9 7 7 ) 763 \ 100 0 0.1- tI i 10 r---l T E a TI * u 0.010- rheology 1.c 57 0 - . P *--. 690 osmosis 0.001- 44 ul S light scattering no 0.1 L / 2 1Tii 1 I 0.1 1.0 9 Fig. 19. Diagram plotted analogously to Fig. 9, showing Gr,Y7/qand pJq forpolystyrene(Mw=8x 106,M,=5.5 x 1O6)intolueneat25"C.Theleft-hand branch of pos/q was determined as in Fig. 9; the right-hand branch comes from the literature 1301 (Mn=2.9x to5). If then in accordance with eq. (13) G,,,/cp is plotted against q, one observes that, for polystyrene with M , z 6 x lo6 in toluene or decalin, Gr,q disappears at rp**x0.02. The critical concentrations following from measurements of elasticity (see Fig. 19 for G,.,+O) correspond well with those from rheology (Fig. 10). Their value is somewhat higher than expected and could be explained by cr,,,<l (see below), but in any case it is far above the critical concentrations measured osmotically or by light scattering (see Figs. 9, 19, and 20). Furthermore, the critical concentrations measured osmotically are inversely proportional to [q] according to Figure 20 and thus contrary to eq. (38). The value corresponds to a state of the solution where thecoilsjust fill the volume of the solution. For spherical coils of diameter h this concentration would be: q* =0.74 x 2.5 1.85 -= - Ctl [sl (39) The values obtained in good solvents follow this relationship very closely. Clearly mere mutual contact between the coils causes disturbances that raise the osmotic pressure above the value calculated by the excluded-volume theory (second virial coefficients dependent on the molecular weight) and, contrary to this theory, make the osmotic pressure independent of the molecular weight. The cause of the disturbances is that at the concentrations in question the coils begin to interpenetrate; during this process entanglements occur between the end fragments of the chains, which may be found mainly in the outer regions of the Such entanglements are not subject to the statistics that led to eq. (9), so that M,, the molecular weight of the network chains, is larger, and q* is smaller, than expected. At this stage the entanglements need not lead to a complete network 764 1'; [I][mlil] 1 0.01 0.001 b 10 1.8 10 + Fig. 20. Critical concentrations c* above which the osmotic pressure po3 or thelight scattering R o is independent of M ,or the viscosity of the solutions rises with q4. plotted against the limiting viscosity [ I T ] for polystyrenes in toluene at 25°C (A, 0 , o), in cyclohexane at 35°C (A), and in decalin (cistrans mixture) at 25°C ( x ) . but may only bind several coils together, thereby considerably restricting the number of possible conformations. The molecules can reduce these disturbances by shrinkage, so long as the increase in free energy due to shrinkage does not exceed the increase due to entanglement. Presumably some shrinkage occurs in the neighborhood of cp*, which in the neighborhood of cp** is largely superseded by stronger entanglement. On shrinkage, the free energy of the N 2 molecules in the network In expression (40) CM is a constant, X K = X , & is a relative expansion factor ( < 1) in the shrunken molecule and xo is the expansion factor ( > 1 ) before the shrinkage, the latter factor depending on interactions with the solvent. From the assumption ai=cp*/cp that is arguable in the vicinity of cp*, the pressure can be calculated as: According to Table 3 this expression agrees sufficiently well with the experimental values for polystyrene in toluene, polyisoprene in toluene, and polystyrene in decalin in the neighborhood of 'p* for zl =0.5. The a,,,< 1 discussed in respect of eq. (38) also supports this hypothesis. Thus for a quantitative interpretation of the experimental pressures the relationship xK= cp*/(p must be assumed, i. e. Angew. Chem. i n r . E d . Engl 16. 751-766 f 1 9 7 7 ) a strong dependence of tl on concentration. As the usually assumed decrease of tl over the whole concentration range to x= 1 in the melt is much smaller, this concentration dependence does not suffice to explain the osmotic pressure. It is possible that entanglements exert their effect even at concentrations below cp* thus influencing the second virial coefficients A 2 of the osmotic pressure. Specifically, it is remarkable that, according to Table 2 and Figure 17 (where it is designated by d/c), the reduced osmotic pressure pos/cp is greater than limp,,/cp by the same percentage, indepenp-10 dently of the solvent efficiency at the same cp/cp*. In fact, theories of the excluded volume do not take into account the conformation restrictions due to contacts between two chains or distant chain fragments, at least not explicitly. The parameter derived from A2 o r from the expansion factor tlo (in dilute solutions) may therefore contain an entropy contribution due to conformational disturbances. 8. Abnormal Effects of Entanglement at Very High Concentrations At concentrations above 20 % conformational changes may be subject to hindrance[l4I, which has not been discussed above. At large entanglement numbers N , the distance of the ends of the network chains of the entanglement network is greater than the distance between spatially neighboring network junctions, in polystyrene for example 1.4 x cm as opposed to 0.45 x 10-6cm (see Fig. 5). Consequently, the m - 1 subchains of a chain cannot be arranged as freely as is assumed in eq. (27). It is not easy to calculate the so it must be estimated114]. Starting from one entanglement (network junction) a network chain ( Z , segments) passes through several (namely n) lattice planes of such disturbing network junctions. The twelve nearest neighbors divide the totality of spatial directions into g groups of directions of almost equal size. The network chain decides in favor of one of them and thus reduces the number of its possible conformations to the fraction l/g. It carries out this limitation n = ( M c / M * ) - 1 times, so that the numerator in eq. (27) must be increased by the factor 9". M* is the molecular weight of a coiled chain fragment whose ends have the same distance as the spatially neighboring network junctions. Reckoning that network junctions acting in this way may include both the entanglement points N , and, to a smaller degree (factor j < I), the n/2 quasi-network points assigned to the n segments of the network chains, we obtain and the distance separating the ends of the chain fragments of molecular weight M* becomes: and for j = 1 : (44b) The number n is then introduced into eq. (31), giving: The bracket at Gr,v in eq. (35) then receives for jn/2 < 1 a term proportional to and for j n / 2 1~ a term that is proportional to c p [ 1 4 ] . As a result, pOs now increases more steeply with cp than the second term in eq. (35) would lead us to expect. In addition, at high cp the first term in eq. (35) has a large influence on pas, independently of the value of the parameter x,. The n portions of the network chains which may disturb the conformational changes of the chains need not also act elastically, at least not at small deformations where the number n is not altered by deformation. Figures 9 and 19 show that at very high concentrations the experimentally determined relaxation modulus has an almost normal dependence on concentration, and thus it is not possible to draw any conclusions about the conformational hindrance considered here from measurements of elasticity. 9. Final Remarks The concept of entanglement of thread molecules is confirmed by many experimental findings and lucidly explains some important properties of melts and concentrated solutions, i. e. of noncrystalline systems. Chain entanglement certainly also occurs in partially crystalline polymers. This explains why polymers that have been stretched, i. e. deformed in the crystalline state, shrink on melting: the sample regains the form that it had before being stretched. The entanglement network has clearly withstood the crystallization and the deformation of the partially crystalline material and on melting appears as a deformed network with a restoring force corresponding to its modulus G , [see eq. (1 3)] and the deformation due to stretching. Some effects of entanglement still remain to be treated more precisely-e. g. those occurring in phase separation, in intramolecularly cross-linked molecules (the effect of crosslinking so far observed for such cases is surprisingly and in the chemical reactivity of the groups and structure components ( e .9. double bonds) situated at entanglement positions. In any event, in discussions of the properties of polymers it should always be borne in mind that entanglement can strongly affect the properties of soft polymers and solutions. Received: 6th May 1966 [A 182 IE] German version: Angew. Chem. 8Y. 773 (1977) Translated by Express Translation Service, London For j < 1, therefore: _ _ _ ~ [t] 4ngm.. C h n . lnt. Ed. Engl. 16. 751-766 ( 1 9 7 7 ) H . G . EILrs: Makromolekule. Hiithig & Wepf. Base1 1972. 765 [2] R. G. Kirste, W A . Kruse, J . Schelten, Makromol. Chem. 162, 299 (1972). [3] a) A . I/: Tobolskg: Properties and Structure of Polymers. Wiley, New York 1960; b) Revised edition by M . Hofmann, Berliner Union, Stuttgart 1967, pp. 118f. [4] W W Grawsfey, Adv. Polym. Sci. 16, 1 (1974). [5] a) F . Btreche: Physical Properties of Polymers. Interscience, New York 1962, p. 61; h) J. Chem. Phys. 20, 1959 (1952); 25, 599 (1956). [6] M . Hofmann, Rheol. Acta 6. 92 (1967). [7] P. J . Flory: Principles of Polymer Chemistry. Cornell Univ. Press, New York 1953, p. 402. [8] H . A . Sruurr: Die Physik der Hochpolymeren. Springer, Berlin 1953, Vol. 2, p. 655. [9] M . Huj>nann, Rheol. Acta 6, 377 (1967). [lo] See ref. 171. p. 405; W Kuhn, F . Griin, Kolloid-Z. 101, 248 (1942): cf. P. J . Flory, Angew. Chem. 87, 787 (1975). [I 11 See ref. [3a], p. 94. [12] M . Hofmann, Kolloid-Z. Z . Polym. 250, 197 (1972). [I31 See ref. 171. p. 425; P . J . F l o r j , 7: G . Fox. J r . , J. Am. Chem. SOC. 73, 1904 (1951): J. Polym. Sci. 5 , 745 (1950). 1141 M . Huflmunn. H . Kriimer, R. Kuhn: Polymeranalytik. Thieme, Stuttgart 1977, Vol. 1 . [l5] M . Hofmann, K . Rother, Makromol. Chem. 80, 95 (1964). [I61 G. Dobrowofski, W Schnabef, Makromolekulares Colloquium, Freiburg, March 3, 1977. [17] M . Hofmann, Makromol. Chem. 153,99 (1972). [18] M . Hoffmann. Rheol. Acta 6, 82 (1967). [!9] M . Mooney. J. Appl. Phys. 1 1 , 582 (1940); R . S. R i d i n , D . $4Saunders, Philos. Trans. R. SOC.London A243.251 (1951). [20] M . Hqffmann, Makromol. Chem. 174. 167 (1973). [21] a) A. S. Lodge, Rheol. Acta 7 , 379 (1968); b) R. Takserman-Krozer, A . Ziabicky, J. Polym. Sci. A2, 7, 2005 (1969); A2, 8, 321 (1970); c) F. S. Edwards, Proc. Phys. SOC.London 85, 613 (1965); 91, 513 (1967); 92, 9 (1967); Discuss. Faraday Soc. 49, 43 (1970); J. Phys. A 6 , 1169, 1186 (1973). 1221 See ref. 171, pp. 577, 512; P. J . Flory, J . Chem. Phys. 18, 108 (1950). 1231 M . L . Huggins, J. Phys. Chem. 46, 151 (1942). [24] M . Dauuil, J . P Cotton, €3. Farnoux, G . Janninck, G . Sarmu, H . Benoit, R. Dupfessiu, C . Picot, P. de Gennes, Macromolecules 8, 804 (1975). [25] P. Debye, J. Phys. Colloid Chem. 51, 18 (1947). 1261 G. K Sehufz, 2. Phys. Chem. 193, 168 (1944). 1271 See ref. [5a], p. 19; P. Debye, F. Bueche, J. Chem. Phys. 11, 470 (1943); A. Ishihora. J. Phys. SOC.Jpn. 5, 201 (1950). 1281 See ref. [7], p. 599. 1291 G. Gee, L . R. G. Peloar, Trans. Faraday SOC.38, 147 (1942); G. Gee, W I . C . Orr, ibid. 42, 507 (1946). 1301 C . E . H . Bawn, R . F . J . Freeman, A . R . Kamallidin, Trans. Faraday SOC.46, 677 (1950). 1311 J . Furukawu, H . Inaguki, Kautsch. Gummi Kunstst. 29.744 (1976). [32] U . Eisele, Lecture at the IISRP Conference Williamsburg (USA) 1976. Flexible Drug Molecules and Dynamic Receptors[**] By R. J. P. Williams[*] When a small flexible drug molecule binds to its likewise mobile receptor (protein, membrane etc.) the shape and function of both can change. The study of the nature and extent of these changes by several independent methods gives an insight into the mode of action of drugs. The static lock and key model will most probably have to be revised or be replaced by the nebulous concept of dynamic states. 1. Introduction The basic problem of drug action in biological systems is easily formulated. In order to interact with a biological system any drug must first bind and may then react, whence a comprehensive minimal two-stage reaction path can be written where D is the drug which undergoes a rapid reversible binding to a biological receptor L, i.e. a protein, DNA, RNA or a membrane etc., giving DL. This reaction has an equilibrium binding constant [*] Prof. Dr. R. J . P. Williams Inorganic Chemistry Laboratory South Parks Road, Oxford OX1 3QR (England) [**] This article is based on the Merck, Sharp and Dohme Scientific Lecture 1976 given in London. It was also given in outline at the Belgian Chemical Society Meeting in Namur, 1976. 766 and there can be many successive steps of this kind before the 'final' binding condition is reached. The second step in the above simplified scheme [eq. (I)], which may or may not be a required part of action, is an irreversible chemical combination of D in the form D' with a part of the biological system L', where L' may be a protein, RNA, DNA, a membrane etc. which has incorporated D (or D'). L and L' could be the same receptor site of course. The reaction rate constant in its simplest form is a first order rate constant, kDL, but again many such steps could be involved. My analysis of drug action will be based upon the structural features of these reaction paths starting from the structures of D and L themselves. By structural features I imply the whole series of conformational states through which the two species D and L must go in order to reach some final state DL or D'L'. Recently the nature of such pathways has been stressed by Feeney, Roberts and Burgen"', who were motivated by their observations, using nuclear magnetic resonance (NMR) spectroscopy, on the mobility of certain drug and hormone molecules. My independent and parallel interest in the problem of conformational mobility has arisen through studies of both small and large molecules in solution also using sophisticated NMR methods for conformational analysis. In the past the solution structures of D and L (which Angrw. Chem. I n t . Ed. Engl. 16, 766-777 i1977 J

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