DILUTE SOLUTION PROPERTIES OF POLYOCTENE ‘X Thesis for ”to Degree of M). D. MECHIGAN STATE UNIVERSITY ' Larry E. Ballard 1963 ' MmHIem STATE umzvsnsm EAST LANSING, MICHIGAN MICHiGAN STATE UNWERSITY w ABSTRACT DILUTE SOLUTION PROPERTIES OF‘POLYOCTENE—l by Larry E. Ballard A sample of atactic polyoctene-l was fractionated and character- ized by light scattering, osmometry, viscometry and, phase equilibria measurements. Three molecular weight—viscosity relationships were established by correlating molecular weights determined by light scattering with viscosity data. They are in cyclohexane at 300C [7]] = 5.75 X 10—5 MO'78 in bromobenzene at 250C [77] = 2.90 x 10‘5 M°°75 in phenyl ethyl ether at SO.bOC [7?] = 6.5h x 10-4 M°°5°. Mean square end-to-end dimensions of the polymer in bromobenzene were calculated from light scattering data by the Zimm method as well as Debye's dissymmetry technique. The universal hydrodynamic parameter E from polymer viscosity theory was calculated and found to agree very well with the commonly accepted value. From phase study data, the theta temperature of the polymer in phenyl ethyl ether was found equal to SO.b°C. A comparison of the ratio of the average end—to-end dimensions to the degree of polymerization for a series of poly a-olefins was made. It was found that the ratios were in the order polystyrene > poly- octene-l > polybutene > polyprOpylene CZ.polyisobutylene. This indicates the dimensions are proportional to the size of the pendant groups. Larry E. Ballard The lower critical solution temperatures for a series of solutions of polyoctene-l fractions in n-pentane were measured. It was found that the precipitation temperature could be related to the molecular weight through the Flory interaction parameter, )(1. A plot of the lower critical solution temperatures against l/kl/Z gave a straight line. The intercept of this line at infinite molecular weight, i.e., i/xl/Z = 0, was defined as e that is, the lower critical L) miscibility temperature for polymer of infinite molecular weight. The 6L for polyoctene-l in n-pentane was found experimentally equal to hBLOK and was calculated from a relationship based on the Prigogine cell model of solutions (assuming )(lc = 0.50) to be b370K. DILUTE SOLUTION PROPERTIES OF‘POLYOCTENE-l By Larry E. Ballard A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF‘PHILOSOPHY Department of Chemistry 1963 ACKNOWLEDGMENTS I wish to express my sincere appreciation to Dr. Jack B. Kinsinger for his patience and assistance during the course of this investigation. I also extend grateful acknowledgment to the Petroleum Research Fund whose fellowship provided financial support during this period. I also wish to eXpress my thanks to my wife, Anna, for her support and for her assistance in preparing the manuscript, and to my parents, I express my gratitude for the many sacrifices they made and for the guidance I received from them which made possible the completion of this phase of my education. ii TABLE OF CONTENTS INTRODUCTION THEORY . . . . . . . . . . . . . . Thermodynamics of Polymer Solutions Molecular Extension . . Phase Study . . . . . . . . Fractionation . . . . . . . Osmometry and the Second Virial Coefficient Viscosity . . . . . . . Light Scattering . . . Lower Critical Solution Temperature EXPERIMENTAL . . . . . . . . . . . . . . Fractionation Phase Studies . . . . . . . . . . . . . Viscosity . . . . . . . . . . . . . . . . Osmometry . . Light Scattering. . Lower Critical Solution Temperature RESULTS . . . . . . . . . . . . . . . Fractionation . . . . . . . . . Phase Studies . . . . . . . . . Light Scattering . . Viscosity . . . . . . . . . Osmometry . . Lower Critical Solution Temperature . DISCUSSION OF‘RESULTS . . SUMMARY . . . . . . . BIBLIOGRAPHY iii Page \ON‘IJZ" 1:" lO 17 21 25 39 39 ho _ h2 113 50 52 Sb 5b 56 68 71 73 78 92 9h Table II. III. IV. VII. VIII. IX. XII. LIST OF TABLES Fractionation data . . . . . . . . . . . . . . . . . . . Thermodynamic parameters from phase equilibria studies. Polyoctene-l in phenetole.. . . . . . . . . . . . Light scattering results in bromobenzene at 25°C . . . Viscosity results for polyoctene-l from fractionations l and 2 . . . . . . . . . . . . . . . . . . . . Osmotic pressure data for polyoctene-l in cyclohexane at 300C 0 O O O O O O O O O O O O O O O 0 O O O I 0 Lower critical solution temperatures for poly a-olefin fractions in n—nonane o o o o o o o 0 o o o o o o o The expansion factor and its dependence on the molecular weight of polyoctene-l in cyclohexane and bromobenzene Values of (Fag/M)1/2 as determined by light scattering . Comparison of (FZC,/'I).1P.)1/2 for atactic poly a-olefins . The thermodynamic interaction parameter as a function of molecular weight and the observed and calculated TCL for polyisobutylene in n—pentane. . . . . . . . The thermodynamic interaction parameter as a function of molecular weight and the observed and calculated TCL for polyoctene-l in n-pentane . . . . . . . . value of 62 obtained by extrapolating T to infinite molecular weight and calculating from equation (11.5) using x = 0.5. . . . . . . . . . . . . . . . . iv Page 55 57 69 7O 71 7h 77 80 85 86 89 91 LIST OF FIGURES Figure . Page 1. Phase diagram of the system n-hexane plus nitrobenzene. . 26 2. Phase diagram of the system diethylamine plus water . . . 26 3. Phase diagram of the system m-toluidine plus glycerol showing upper and lower critical temperatures with L.C.S.Tk < U.C.S.T. . . . . . . . . . . . . . . . . . 26 h. Phase diagram of polymer-solvent system showing upper and lower critical temperatures with L.C.S.T. >IU.C.S.T.. 26 5. Variation of chemical potential of n-hexane as a function of nitrobenzene concentration . . . . . . . . . . . . 28 6. Variation of molar free energy with composition . . . . . 29 7. Molar enthalpy as a function of composition . . . . . . . 32 8. Thermodynamic excess functions in the neighborhood of an U.C.S.T. . . . . . . . . . . . . . . . . . . . . 33 9. Thermodynamic excess functions in the neighborhood of a L.C.S.T. . . . . . . . . . . . . . . . . . . . . . 33 10. Variation of'FB starting from 0 to L.C.S.T. and to U.C.S.T. . . . . . . . . . . . . . . . . . . . . . . 35 110 A = O o o o o o o o o o o o o o o o o o o o o o o o o o 38 12. hAB < (1/2 R)2 . . . . . . . . . . . . . . . . . . . . . 38 13. LAB > (1/2 R)2 . . . . . . . . . . . . . . . . . . . . . 38 1h. Plot of log [77] in cyclohexane against log [7?] in bromobenzene . . . . . . . . . hh 15. Plot of log Ah vs. time to determine time constant (0). Plot of Ah vs. time to show diffusion through memrane (A) O O O O O O O I O O O O O O O O O O O O h? 16. Osmotic equilibration curve at 30°C for oLyoctene-l, FlOA in cyclohexane. C = 0.80h7 gm 100 cc. . . . . . N9 17. Binary phase diagrams for polyoctene- 1 in phenyl ethyl ether . . . . . . . . . . . . . . 58 LIST or FIGURES (Cont.) Figure Page 18. 1/TC for polyoctene-l fractions . . . . . . . . . . . . . 59 19. Zimm plot for polyoctene-l fraction 5A in bromobenzene at 250C 0 O O C C O C O O C O O C O C C C O O C C O O 61 20. Zimm plot for polyoctene-l fraction 6A in bromobenzene at 250C 0 O C O C O C C O C C O C C C C C C O O C O C 62 21. Zimm plot for polyoctene-l fraction 7A in bromobenzene at 25°C 0 O O O O O O O O O O O O O O O O O O O O O O 63 22. Zimm plot for polyoctene-l fraction 8A in bromobenzene at 250C 0 O O O O O C O O O O O O O O O O O I O O O O 6h 23. Zimm plot for polyoctene-l fraction 9A in bromobenzene at 250C 0 O O O O O O O O O O O O O O O O O O O O O O 65 2h. Zimm plot for polyoctene-l fraction 10A in bromobenzene at 250C 0 O O C I O C C C C O O C C C C C O O O O O O 66 25. Intrinsic viscosity-mplecular weight relationship for polyoctene-l in (1) cyclohexane at 30°C 0 , (2) bromobenzene at 25°C A and (3) phenetole at 50.h°C 0. 72 .26. Osmotic pressure vs. concentration for polyoctene-l in cyclohexane at 30°C . . . . . . . . . . . . . . . . . 7h 27. Phase diagrams for L.C.S.T. of polyoctene-l fractions in n‘pentane o o o o o o o o o o o o o o o o o o o 0 7S 28. Lower critical solution phase diagrams; T‘ vs.'wt.‘ . CL fraction polymer . . . . . . . . . . . . . . . . . . 76 29. Viscosity plot for polyoctene-l according to equation (1002) O O O O O O O O O O O O O O O O O 0 O O O O 0 82 30. Root-mean-square end-to-end dimensions of polyoctene-l as a function of the molecular weight . . . . . . . . 8b 31. A plot of the lower critical solution temperature against the molecular size function, 1/k1/i, for two poly a- olefins in n-pentane. PoLyoctene-l (CD); polyisobutyl- ene (A) ref. 8 and ([3) ref. 7. . . . . . . . . . . . 88 vi INTRODUCTION Since it was first demonstrated by Nattal that stereoregular a-ole- fins could be prepared by Specific catalysis, a number of results have been published-2:3:4 which indicate a dependence of certain physical measurements on the polymer configuration, i.e. the distribution of : asymmetric centers. The present work is a continuation of a program, being carried on primarily in this laboratory, designed to study the re- lationship between physical properties of a-olefin polymers and their chain configuration. The present work was carried out with the purpose of characterizing a sample of atactic polyoctene-l and thereby adding to the data which are available for the poly a—olefins in general. In addition, polyoctene-l is of interest because of its relatively large pendant group. One object of this study is to discover how the pendant group will effect such measurable quantities as the average end-to-end length of the molecule. At first thought, it would be ex- pected that the end-to—end length of a polymer would be directly pro— portional to the length of its pendant groups. That is, a long pen- dant group would be more apt to interfere with other groups and the polymer backbone and thus have the effect of expanding the polymer. However, Chinai and collaborators5:6 have investigated a homologous series of n-alkyl methacrylate polymers and found that the ratio of the end-to-end dimensions of these polymers to the degree of polymerization fell off in the order n-hexyl >in-octyl > methyl > ethyl > n-butyl. This order is curious in that the size of the polymer is not dependent on the size of the pendant group. In like manner, the end-to-end dimensions measured for the polyoctene-l will be compared with the data of other 1 2 a-olefins and any apparent order or anomalies with reSpect to the length of the side groups will be noted. Aside from the standard characterization procedures, measurements of the lower critical solution temperature (L.C.S.T.) of polyoctene-l, as well as several other a-olefins in n-pentane were undertaken. The fact that polymers will precipitate from solution when the solution is heated sufficiently high has only recently been recognized7:5. It ap- pears that this phenomena is related to the approach of the solvent to its critical temperature and is associated with a rapid decrease in the entropy of mixing and a negative heat of mixing. It is believed that this is a universal phenomena characterizing all polymer-solvent systems. Several workers725 have determined the L.C.S.T. for polymer-solvent systems and found that the L.C.S.T. decreases with increase in molecular weight. Also the L.C.S.T. was related theoretically to the Flory inter- action parameter )(1 and the solvent molecule Size9. However, no work has been carried out in which polymer fractions have been studied. It was therefore of intereSt to determine the L.C.S.T. of solutions of characterized polymer fractions and to relate the L.C.S.T. to polymer molecular weight. Upon considering the relationship between molecular weight and L.C.S.T., a new parameter is immediately suggested. This new parameter, which shall be referred to as 0L, is analogous to the Flory 0U which is the upper critical miscibility temperature for infinite molecular weight polymer. 0n the same basis,_9 is the lower critical miscibility temperature for, L infinite molecular weight polymer. That is, below the temperature 9L (provided we are at the same time above GU), polymer of any molecular weight is miscible with the solvent. Since no exact theory for this 3 phenomena is at present available, 0L will be determined in a manner similar to that used to find 0U. However, any relationship between molecular weight and the L.C.S.T. which is linear could be extrapolated to infinite molecular weight to find 0L. THEORY Thermodynamics of Polymer Solutions Because of the size and conformations of dissolved polymer molecules, their solution behavior is considerably different from that of low molec- ular weight substances. Thus special theoretical treatment is required to explain their solution properties. Even at low concentrations, poly- mer solutions exhibit large deviations from ideal thermodynamic behavior. These deviations from ideality arise largely from very high entropies of mixing as a result of the large difference in molecular Size between the polymer and the solvent. By a statistical mechanical treatment in which polymer segments and solvent molecules are allowed to occupy sites in a lattice, Flory1° and Huggins11 independently were able to Show that the conformational entropy of mixing was given by AS = -k(nl ln/U'l + 1'12 III/v.2) (1.1) M where/V'l and/mfg are the volume fraction and n, and n2 are the num— ber of molecules of solvent and solute and k is the Boltzmann constant. The heat of mixing can be seen to originate in the replacement of some of the contacts between like species in the pure solvent or poly- mer with contacts between unlike Species in the solution. According to the lattice theory with each cell able to accommodate either a solvent molecule or a polymer segment, three types of first neighbor contacts are possible. That is, it is possible to have a solvent-solvent, a sol- vent-polymer, or a polymer-polymer contact. If the energies associated with these contacts are represented by ”11: “12: and "22 respectively, the change in energy for the formation of an unlike contact pair can be represented by 5 Aw12 = ”12 - (l/2)(w11 + "22) (1.2) Following the van' Laar' treatment for the heat of mixing, we can write AHM = ZAW1zninfe (1'3) This equation can be written as AHM ‘ kTX1n1/U’2 (1.14) where we define X1 = Zimm/RT (1-5) as a dimensionless parameter which characterizes the interaction energy between polymer and solvent, 2 is the coordination number of a polymer segment, and n1 is the number of solvent molecules. If it is assumed that Aw12 is independent of temperature, that is, if Awlz contains no entropy contribution and the only entropy contri- bution is from the conformational entropy as given by equation (1.1), one can find the free energy of mixing from equations (1.1) and(l.h). That is, ARM = AHM - TASM = kT[nl lnnfl + n2 lnAfz + )(1D1fl721 (1.6) From this expression it is possible to derive several useful relation— ships involving experimentally obtainable quantities. For example, if (1.6) is differentiated with respect to n1 (realizing thatlnrl and/172 are functions of n1 and n2), we can find the relative partial molar free energy, AF, (i.e., the chemical potential per mole of the solvent in the solution). Thus 0 AFl = “1 ‘ H1 RT[1n(l-/V2)+(1—1/X)flfz + XanZP-l (1.7) RT 1n a1 (1.8) where x is the number of chain segments per polymer molecule and is given by the ratio of the molar volumes of the solute and the solvent, and a1 is the activity of component 1. 6 In like manner, if we differentiate (1.6) with respect to n2, we obtain for the solute AF“2 = 112 - 112° RT[ 1n/zr2 - (x-1)(1-/U'2) + X1x(l-nf2)2] (1.9) RT 1n a2 (1.10) The theory just discussed is based on the assumption that there is a uniform concentration of polymer throughout the solution. It is more likely, however, that in dilute solutions individual polymer molecules become separated from each other by regions of pure solvent and the con- centration of polymer segments must become non—uniform in the solution. This problem is approached by deriving a general expression for the total free energy of interaction between all of the segments in a volume ele- ment, (5V, which is considered small enough so that the expected seg- ment density may be considered to be the same for all portions of 6V. ' It has been shown that the free energy of mixing polymer segments with solvent in the volume element (5V is given by 6(AFM) .. hflén, 1n(l-n}'2) +X16n1m’2] (1.11) whereilrz refers to the volume fraction of the polymer in (5V. The chemical potential of the solvent in 6V is given by (u. - 111°) = RTtan-w‘z) "0‘2 + X10521 (1-12) If (1.12) is expanded in series and if powers oftlfé larger than the second are neglected, we have (u. - 11.0) = min/2 - X1W'221 (1.13) These expressions may be regarded as the excess chemical potential re- sulting from non-ideal contributions. Equation (1.13) can be written in the form (11. - 11.0) = RuK, 41/9/2122 (1.11.) where f<1 and UV, are heat and entropy parameters defined as 7 AH, %.RTK1n/’22 (1.15) 15, = 1251/,ng (1.16) Comparing equations (1.13) and (1.1h), it is seen that K1 - $01 = Xi - l/2 (1.17) It is sometimes preferred to use as a parameter the ideal temperature defined by e = Kl'r/SL/l (1.18) so that W1 ' K1 = W1(1-9/T) (1.19) The excess chemical potential can be written (11, - (1,0) = -RTy/,(1 — e/T)/U’22 (1.20) In a thermodynamically poor solvent, i.e. where both K1 and W1 are positive, 9 will also be positive. When T = 9, the chemical potential due to segment-solvent interactions is zero according to equation (1.20). Thus at T = e, the excess chemical potential is zero and deviations from ideality vanish and the molecules can interpenetrate one another freely with no net interaction. At temperatures below 0, they attract one another, and at temperatures much below 9, precipitation of the polymer occurs. It will be seen later that G is the critical miscibil- ity temperature for polymers of infinite molecular weight. Molecular Extension”a In general, each molecule in a dilute solution will tend to exclude all others from the volume which it occupies. This leads to the concept of an excluded volume from which a polymer molecule effectively excluded all others. If this concept of excluded volume is extended to individual molecules, we find that, because of the obvious requirement that two 8 segments cannot occupy the same space, the chain will extend over a larger volume than would be calculated theoretically. The extended conformation of a polymer molecule is determined by an equilibrium~between an expan- sion force due to the excluded volume and a contraction force due to a distortion of the molecule beyond its most probable conformation as de- termined by bond length and valence and internal rotational angles. The distorted chain may be considered to be expanded by a factor a over the unperturbed dimensions (the dimensions determined by the bond angles and the lengths only) of the molecule. Thus the actual root-mean-square end-to-end length of the molecule, F2, is given byaFZo where F30 is the unperturbed dimension determined experimentally from viscosity measure- ments. The total free energy of mixing the polymer segments of a molecule with a solvent consists of the sum of the AFMj for each volume element plus a term AFel for the free energy change associated with alteration in molecular conformation. Thus AF = :AFMJ + are, (2.1) .1 At equilibrium (BAP/Gan, P = o (2.2) This leads to an evaluation of a given by a5 - a3 = 2CMyj1(1-G/T)M1/2 (2.3) where /2fl3/ /2 5 2 _ '3 C, = (27/2 >(fi72/Nv.>(r2./M> (2.1) and a? is the specific volume of the polymer, v1 its molar volume, and N is the number of particles. Equation (2.3) predicts that a should increase without limit as molecular weight increases. Thus (Fay/2511mm 9 increase more rapidly than in proportion to the square root of the molecular weight. This follows from the theory of random chain config- uration which shows that the unperturbed root-mean-square distance ( F20)1/2 is prOportional to Ml/Z, whereas ?2 = aFZO. Also a depends on the intensity of the thermodynamic interaction as eXpressed byl%&(l-e/T) which is equal to Sfll - AC1. The larger this factor, the greater the value of a for a given M. Hence, the better the solvent, i.e. the larger the 90,, the greater the swelling of the 3 molecule. At T = 0 in a poor solvent, a5 - a3 = 0 and a must equal unity. Therefore, at T = 0, the molecular dimensions are unperturbed by inter- molecular interaction and the excluded volume will equal zero at this temperature. 12b Phase Study When the temperature of a polymer solution is lowered, the solvent becomes thermodynamically poorer. Finally, a temperature is reached _ below which the polymer and solvent are no longer miscible in all pro- portions. At this and any lower temperature, a mixture of polymer and solvent will separate into two phases. The condition for equilibrium between two phases in a binary sys- tem is that the partial molar free energy of each component be equal in each phase. Application of this condition to the partial molar free energy given by equation (1.7) gives for the critical concentration at which phase separation first appears fUzc = 1/(1 + x 1/2) ””V 1/x1/2 (3.1) 10 The critical value of )(1 is given by 1 2 1 2 X“, = (1 + x / )2/2x c: 1/2 + l/x / (3.2) Thus the critical value of 9!, exceeds 1/2 by a small increment depend- ing on the molecular weight and at infinite molecular weight equals 1/2. The temperature at which phase separation begins is given by 1/1‘C = 1/e[1 + (1/g[/,)(1/x1/2 _ 1 Thus l/TC(°K) 1 should vary linearly with (l/x + 1/2x)] (3.3) 2 / + l/2x). The theta temperature is seen to be the critical miscibility temperature in the limit of infinite molecular weight. Fractionation Most polymerization procedures yield products which are hetero— geneous with respect to molecular weight. Before the polymer is studied, it is desirable to separate the whole polymer into parts or fractions having a relatively narrow molecular weight distribution. All of the methods used to fractionate polymers are based on the difference in solubility of the different Species present. That is, the solubility, which is related to the chemical potential of the Species in solution, decreases with increase in molecular weight. Thus if the solubility of a solvent is decreased by cooling a solution of a hetero- geneous polymer, a point is reached when phase separation occurs and two layers form. The system will then consist of a dilute phase with low polymer concentration and a precipitated phase w ith high concen- tration. Because of the lower solubility of the higher molecular weight species, they will be contained mostly in the precipitated phase, and the more soluble lower molecular weight species will remain in the dilute solution phase. 11 Likewise if the solubility of a solvent is decreased by adding a non-solvent to it, a point is reached when enough non-solvent is added to cause the high molecular weight species to precipitate. If this precipitated polymer is removed and additional non-solvent is added, the polymer of slightly lower molecular weight will precipitate. Thus a scheme is offered whereby a whole polymer can be separated into frac— tions of different molecular weight. Osmometry and the Second Virial Coefficient An osmometer is a device in which a solution is separated from pure solvent by a membrane permeable only to solvent molecules. The activity of the solvent in the solution is less than that of the pure solvent, and if the system is to be kept in equilibrium, the activity of the solvent on both sides of the membrane must be brought into equi- librium. This may be done by applying an excess pressure to the solu- tion side, either mechanically or by developing a hydrostatic head. The excess preSsure required to reach equilibrium is called the osmotic pressure, n, and the change in activity with pressure is given by the equation (aln al/aP)T N = vl/RT (11.1) 3 where V, is the molar volume of the solvent. Thus at osmotic equilibrium, 1 n f a 1n a1 = f (VI/RT) ap (1.2) o 0 Since v1 is essentially independent of pressure, we can write - ln a1 = n vl/RT (11.3) If the solution is sufficiently dilute, a1 = N1 where N1 is the mole l2 fraction of the solvent, and since N1 is near unity, - ln N1 261 —N1’/"vN2 xcvl/M (11.11) where c and M are the concentration and molecular weight reSpectively of the solute. Substituting these expressions in equation (h.3), we obtain n/c e: RT/M (1.5) However, this expression holds only for ideal solutions and in order to treat polymer solutions, it is necessary to use the thermodynamic relationships discussed previously. According to equation (1.8), we have for a polymer solution RT 1h a1 = ln(l-/U'2) + (1 - 1/x)nf,_ + le'zz (11.6) Substituting (b.6) directly into (h.3), we have n = -(RT/v1)[ln(l-nf2) + (1 - l/x)/U’2 + Xlrv'zzl (11.7) If the logarithmic term is eXpanded and only terms in powers of Va of order three or less are retained, then = (RT/v.)W2/x + (1/2 - .X1W'22 Mfg/3 + ...J (11.8) chwhere A7 is the (partial) specific volume of the polymer 2 Since ”2 and since x is the ratio of the molar volume of polymer and solvent, we have n/E/xvl = efif/kv, = c/M (h.9) Thus fi/c = RT/M + RT(nI2/v,)(1/2 - X1) 0 + RT(fl73/3v1)c2 + (11.10) The first term on the right is the van't Hoff ideal term. At infinite dilution fl/c must approach this limit. The higher order terms repre- sent the deviation from ideality. It has already been pointed out that the lattice model theory suffers from the invalid assumption that the interaction of the segments 13 of a given polymer molecule with the segments of all other polymer mol- ecules are the same as would be expected if theSe latter segments were randomly distributed over the entire volume. It is much more likely that at high dilution, the solution may be considered to consist of two more or less distinct regions; i.e., one containing clfisters of polymer molecules and the other region consisting of pure solvent. The expressions for osmotic pressure derived from the lattice model theory are therefore invalid at high dilution. The deviation of a polymer solution from ideal behavior may be thought to arise from interaction of molecules in the solution. Each molecule in a dilute solution of a good solvent will tend to interact with those in its vicinity and thus exclude them from the volume which it occupies. This excluded volume is calculated by considering the interaction between a pair of molecules in solution. Flory and Krig- baum13 derived a general expression for the free energy of mixing in terms of the excluded volume. Thus AFM £5 - nsz[ln V - (u/2)(n2/V)] + Constant (b.11) where u is the excluded volume. The excluded volume u can be expressed by the following relationship: u = 2801(1 - e/T)(fi2m2/v,) F(J £3) (11.12) where m is the weight of a polymer molecule, V1 is the volume of the solvent, and J has the value J = w,(1_e/T)W2/v,) ' (1.13) The value of (33 is given by £3 = (33/21/2113“ 2) Ni’leg/Mfi/z M'Vzas (1.11) where NA is Avogadro's number. The function F(Jg33) is of the nature unis 0. «1|: at K 11 that it decreases as the agrument increases. If we examine equation (1.13), we see that when T = 0, J vanishes. When J vanishes, F(J§3) becomes unity. The dependence of the excluded volume on temperature is seen by examining equation (1.12). AS the solvent becomes poorer by decreasing the temperature, the excluded volume decreases and at T = 6 it vanishes. From equation (1.11) we can derive an.expression for osmotic pres- sure by use of standard thermodynamic operations. Thus n/t = RT[l/M+ (NAu/2M2)c] (1.15) This equation holds only at very high dilution since only binary inter- action between molecules was considered. If the osmotic pressure is expressed as a power series in concentration analogous to the virial eXpansion for a gas, we may adopt the convention form n = RT[A1c + Azc2 4- A3c3 + ...] (1.16) where c expresses the concentration of polymer in gm per unit volume. Recalling that nZ/V = cNA/M and comparing equations (1.15) and 1.16) it follows that 1/M (1.17) NAu/2M2 (1.18) A1 A2 or from equation (1.12) we can show that A. = Wynn/ml - 6/1“) my) (1.19) = JF(Jg33) (1.20) When equation (1.16) is compared with the virial expansion of PV in powers of l/V for a real gas, the analogy is at once obvious. Thus if the gas molecules are regarded as point particles which exert no forces on each other so that u = 0, the second and higher virial coef- ficients, A2, A3, etc. vanish and the gas behaves ideally. Likewise ) . 15 in a polymer solution when T = 0, u = 0, and equations (1.15) and(1.l6) reduce to the ideal van't Hoff's law fl/t RT/M (1.21) which rearranges to II 11V, anT (1.22) and is a direct analogy to the perfect gas law. Thus the temperature T = B for a polymer solution is seen to be the analog of the Boyle point of a real gas, that is, the temperature at which a real gas obeys the relation PV = nRT except for terms in the square and higher powers of l/V. The value of A3 depends on interactions involving three molecules. In order to evaluate A3, it is necessary to consider the region of Space from which the center of gravity of molecule k is repelled by both of the molecules 1 and m. An accurate evaluation of the intergals result- ing from considerations of the problem are difficult and so approxima- tion procedures are used. In one model, the molecules are treated as inpenetrable non-interacting spheres. The results of these calculations show that A. = (5/8>A.2M (1.23) A more detailed analysis14 shows that the numerical coefficient 5/8 should be replaced for polymer molecules by a slowly increasing function of A2. It is less than 5/8 and vanishes as A2 goes to zero. Calling this function g, we have A3 = 9822M (1.21) For many purposes it is preferable to use instead of equation (1.16) the series expansion fl/c =(fl/t)o[1 + [EC + rgcz + ...] (1.25) 16 where B. = Az/Ai (1.26) F. = 13/11, (1.27) and (Ir/ch) = RT/M (1.28) An important consequence of the relationship between the second and third virial coefficient is that the contribution of the third virial coefficient decreases rapidly as the second is made smaller, that is, when poorer solvents are used. It follows that the decrease in slope for poorer solvents should be accompanied by a rapid decrease in curva— ture. Osmotic pressure measurements are made to determine both the molec- ular weight of a polymer and the polymer-solvent interaction. These two values require that the first two terms in equation (1.25) be evaluated. Since the third term is important only as it aids in the accurate evalu- ation of the others, its value can be approximated, and since it makes a negligible contribution in poor solvents, it is sufficient to take for 9 its value in a good solvent, that is 0.2514, and to treat it as a constant15. Thus if g = 0.25 is used and higher order terms are neglected we can write equation (1.25) as n c = (n/t).[1 + ( F}/?)c]2 (1.29) The osmotic pressure data is then treated by plotting (n/c)1/2 against c. The molecular weight is calculated from the intercept of the line and.A2 is determined from the slope and the intercept of the line. If we examine equation (1.2)), we see that A2 is given by the molec- ular weight independent factor J, multiplied by the factor F(Jg33) which decreases slowly from unity to approach zero asymptotically as the 17 molecular weight becomes larger. We can compute A2 easily from the theoretical expression using experimentally obtained parameters. Thus J = (Elm - e/T)'n';"2/v, (1.30) is obtained easily by using values of 1 and 0 obtained from phase studies, and 572 and V, are found from density measurements. The fac- tor £33 can be computed using values of a obtained from viscosity stud- ies. The factor F(Jg33) can be found by successive approximation calc- ulations. Also this factor has been expressed in closed form by Flory, Krigbaum, and Orofinol5217, and by Cassassa and Marrorityla. The calc- ulated values of A2 can then be compared with those obtained from osmometry or light scattering measurements. Viscosity A pr0perty common to all high polymers is that when dissolved, the viscosities of the resulting solutions are considerably larger than those of the pure solvents. This ability to produce a viscous solution is a property of polymers related to the voluminous nature of the randomly coiled chain molecules and So is basically a measure of the Size or ex- tension in space of the polymer molecule. The viscosity of a liquid is commonly determined by timing the flow of a known volume of the liquid through a capillary (viscometer) and is calculated using the relation 7']: Apt — B/O/t (5.1) . where A and B are viscometer constantS,/4)is the density of the liquid, and t is the flow time. The factor B is used to correct for kinetic energy-effects. If t is large and if well-designed viscometers are used, the kinetic energy effect is small and can be neglected. If the kinetic 18 energy effects are small, the viscosities of the solvent and solution, YLand no, can be replaced by the flow times, t and to, since only the relative values are required. (The subscript 0 refers to solvent flow time.) It is customary in polymer science to represent polymer solutions concentrations, c, in gm./100 m1. On this basis, if the viscosity of the solution is 77 and that of the solvent is 770: we can define several useful quantities. If the viscosity of the solution is divided by the viscosity of the solvent, we obtain the relative viscosity, 72rel° The Specific viscosity is defined as 773p = 77rel - l and expresses the in- cremental viscosity attributable to the solute. The ratio 77Sp/c, the viscosity number (or reduced specific viscosity), is a measure of the Specific capacity of the polymer to increase the relative viscosity. The limiting value of this ratio at infinite dilution is called the intrinsic viscosity19 which is designated by [7?]. Thus in mathematical language [721 = (mp/c), _, O = [07,81 - we], _,_ O (5.2) When the concentration c is expressed in gm./100 ml., the intrinsic. viscosity is given in the reciprocal of this unit, deciliters per gm. Intrinsic viscosity may aIso be defined as the following: , [721 = [nanny/c], _,O (5.3) Huggins19 found, for a series of polymer fractions at different concentrations in the same solvent at the same temperatures, that the slopes of the linear portions of the plots of 77sp/c against c were proportional to the square of the intercept, and he proposed the empir- ical relationship 19 = 2 USP/c [771+ k' [77] c (5.1) In this equation, k' is constant for a given polymer-solvent system pro- vided the polymer is pure and homogeneous. It is generally in the range of 0.30 to 0.10. Another relation, due to Kraemerzo, similar to (5.1) is = _ 2 (1177mm [771 mm c (5.5) If the left side of equation (5.1) is expanded in terms of 77Sp/c, we obtain 2 ' 3 [1n(nrel)/c1/c = 332 - 31.2 52,... 77,2 §2 - (5.6) c c c from which we have 11 [l ( )/1 = 1' ( /) (5.7) C_:Onnrelc C_1_;n_o7?SpC If equation (5.1) is substituted into equation (5.6) and powers of c higher than the first are neglected, it is seen that k' + k" = 0.50. This furnishes a convenient aid for extrapolating viscosity data. Usually [TI] is found by plotting both , ”Sp/C against c and ln(7?rel)/c against c on the same graph. This procedure facilitates extrapolation since both intersect the ordinate at zero concentration. If log [7?] of a series of polymer fractions is plotted against log M, a general empfiical relationship, known as the Mark-Houwink equa- tion21, is obtained. The relationship may be eXpressed by the equation in] = 11' Ma (5.8) where K' and a are constants for a given polymer, solvent, and temper- ature. The value of the constant a, when applied to measurements of flexible chains, usually lies between 0.50 and 0.80. Although equation (5.8) is empirical in origin, it can be closely approximated by more complex theoretical expressions. 20 It has been shown that the viscosity measurement gives a viscosity average molecular weight22 defined by the equation Mv=[Z ”iMa]l/a = [2 NiMiHa/Z “811/:11 (5'9) i=1 i=1 i=1 If fractionated polymers are available so that Manv’xMw, any absolute molecular weight measurement can be used with values of intrinsic vis- cosity to evaluate the constants in equation (5.8). However, since MV is nearer to Mw than to Mn’ weight average molecular weights are prefer- red for determining the constants. The theories of the frictional properties of polymers in solution conclude that the intrinsic viscosity is proportional to the effective hydrodynamic volume of the molecule in solution divided by the molecular weightlzc. The effective volume is shown to be proportional to the cube of a linear dimension of the polymer chain. When (F2)1/2 is chosen as the linear parameter, it is shown that12C [7?] = 1 (Rf/2m (5.10 where Q is a universal constant independent of solvent temperature and polymer. We can separate (301/2 into the factors a(F§)1/é, and since (F0)1 2/M is independent of the solvent and molecular weight, we can write [771 = “ROADS/2 111/ch = mil/2:13 (5.11) where K = D (Poms/2 (5.12) which is constant for a given polymer and independent of solvent and molecular weight. It was shown previously that at T = G, a = 1. There- fore at T = 0 we have 21 [7?]. = KMl/Z (5.13) The constant A can be calculated from equation (5.10) if F2 has been determined from light scattering measurements or by some other means and if is measured in the same solvent at the same temperature at which the light scattering measurements were made. From the value of E, (FQ/M)l/2 can be calculated from (5.10) and compared with similar values calculated, assuming free rotation about the bonds. Thus it is possible to obtain information on the effect of hindered rotation about bonds and other perturbing effects, such as side groups on the main polymer chain. Light Scattering When a beam of light passes through a non-absorbing liquid, it is found that the medium is not perfectly transparent but scatters a small amount of the incident radiation. This scattering in pure liquids was shown by Einstein23 to be related to local thermal fluctuations in the density of the liquid which made it optically inhomogeneous. If the inhomogeneity of the medium is increased by adding a solute, the scat- tering intensity increaSes. The intensity of the light Scattered from a solution depends on the polarizability of the molecules compared with that of the solvent, the size of the molecules, and the concentration of molecules. The in- tensity of the light scattered in a given direction from a single mole- cule is proportional to the square of its size or molecular weight. This means, for example, that the light scattered from one molecule of molec- ular weight 2M will be greater by a factor of 2 than the light scattered 22 by two molecules of molecular weight M. This dependence of scattered intensity per particle on a power of molecular weight greater than the first make possible the determination of molecular weight and polymer size. The problem of measuring molecular weights of polymers by light scattering therefore becomes a matter of determining the intensity of scattered radiation. The intensity of scattered light is expressed in terms of the turbidity 2', defined as the fraction of the light scattered in all directions from the primary beam per cm. of path. The turbidity is measured experimentally by comparing the scattered intensity at a given angle to the incident intensity of the light. Debye34 related molecular weight to scattered intensity by the expression 3.3% = 117.1 ‘1' 2A2C (6.1) where H = 32w: no2 (92 2 (6 2) 3NA 7.71- dc ’ is constant for a given polymer-solvent system and a specific wave length of light, I. The other constants are Avogadro's number NA’ the refractive index of the solvent no. The (dn/dc) is the refractive index increment of the solution due to dissolved polymer. This is a measur- able quantity which is pr0portional to the excess polarizability of the molecules in the solvent. A2 is the second virial coefficient and c is the concentration of the polymer expressed in terms of weight per volume. Equation (6.1) is satisfactory for particles smaller than about (l/20)A. For larger particles, interference occurs in the scattering pattern and the scattering enve10pe becomes unsymmetrical. In this 23 case, it is necessary to correct equation (6.1) with a factor P(0) which is a shape factor that corrects the observed scattering to that which it would be if there were no interference. Thus equation (6.1) becomes %E = METBT + 2A2c (6.3) Zimm was able to Show25 that the shape factor for randomly coiled polymer molecules (that is, those whose radial segmental density distri— bution can be represented by a Gaussian function) may be represented approximately as _ 3- LimPl(O) = 1 + g},— r2 31.12-2- + (6.1) C-9O where sin 0 is the viewing angle as measured from the direction of the incident light. Combining (6.1) with equation (6.3), we have 2— 1.1111 Liz—C: = W-fi- (IT’S—:31”? 51112-2"? ...) (6.5) c a> 0 If the dissymmetry method of Debye is used to calculate the molec- ular weights and sizes, it is necessary to measure the scattering inten- sity, ie, at two angles symmetrical about 90°, for example, at 15° and 135°. The dissymmetry coefficient, 2 = i90-45/190+45, thus obtained can be used to calculate the value of (172)1 2 for the molecules being measured. We can then use (F?)1/é with equation (6.1) to calculate the shape factor, P(G), and then use P(G) in equation (6.3) to calculate the molecular weight. If the dissymmetry is large, the scattered intensity, and therefore the turbidity, will be functions of the angle at which it is determined, as well as the concentration. Zimmz6 solved this problem by plotting the data simultaneously as a function of angle and concentration. Thus Hc/qf in equation (6.3) is plotted against sinZO + kc (k is an arbitrary (J 9‘. A .-1U ~'L‘f‘ '.,C 4‘ *1 .flnvf‘y: “Will..- x (/2 (U 21 constant chosen so that kc assinz g ) and the curves are extrapolated both to zero angle and zero concentration. These limiting values are then extrapolated to the common intercept at zero. The limiting slope of the zero angle line yields the second virial coefficient, A2; the ratio of the limiting slope of the zero concentration line to the inter- cept gives the mean-square end-to-end dimension of the molecule, and the intercept gives the reciprocal of the weight average molecular weight, Mg. The dimensions of the molecules obtained in this way are the 2- average values, and some knowledge of the molecular weight distribution is required in order to convert them to weight average values. Many molecular weight distributions can be approximated by a function of the form h+l h _ M f(M) = lfi—r- M e y (6.6) where h is a parameter characterizing polydispersity. Zimm has shown“:26 that for such a distribution, h is given by h = (fi;/fin - 1)‘1 (6.7) and that Mz =1») = E (6.8) h+2 h+1 h If we assume that a polymer has this particular distribution, we can convert the z-average dimensions to the weight average dimensions by use of the relation r = rw(h + 2)/(h + 1) (6.9) Z 25 Lower Critical Solution Temperature If two unlike liquids are mixed, two results are possible. The liquids will either dissolve each other completely to form a single phase or the liquids will not dissolve each other completely and two phases (each of which is a solution) will form. If the homogeneous mixture of two liquids is not miscible at all temperatures and at all compositions, then at some other temperature or composition, the two will separate into two phases. For example, n-hexane and nitrobenzene are miscible in all proportions, if the temperature is above 19°C.. Likewise, outside the limits of about 0.15 and 0.80 mole fraction nitro- benzene, the two liquids will be miscible at all temperatures. If we plot the phase separation temperature against the mole frac- tion of nitrobenzene, the results will be as shown in Figure 1.? At the point C, the maximum in the curve, the two liquid layers become identical and the liquids will be miscible in all proportions. This point is called the upper critical solution temperature (U.C.S.T.) and also called the upper critical consolute temperature. In other two component systems, a different type of behavior may be observed. For example, in the system diethylamine and water, shown in Figure 2, there exists a temperature below which the liquids are miscible in all pr0portions. This point represents a lower critical solution temperature (L.C.S.T.). Finally, there are liquid systems which exhibit both an upper and a lower critical solution temperature. An example of this is the system n-toluidine and glycerol as depicted in Figure 3. In this particular case, the two phase region is a closed 100p and the L.C.S.T. is lower than the U.C.S.T. In another case, the 26 One Phase 20 15 T°C 10 L 1, B 0.50 0.75 1.0 vw—A J O 0.25 x1 nitrobenzene Fig. 1.27a Phase diagram of the system n-hexane plus nitrobenzene. 125 ‘— -'_'-‘- Thu 100 h 75 r- Two Phases T°C 50 P 25 +- 0 l ' L - .1 - - ‘— TC]— 0 0.25 0.50 0.75 1.0 x glycerol Fig. 3.27a Phase diagram of the system m-toluidine plus glycerol showing upper and lower critical temper- atures with L.C.S.T. U.C.S.T. 2? L.C.S.T. is higher than the U.C.S.T. and there are two, two phase regions separated by a one phase region. This type of behavior has been obser- ved for polymer systems and is the one which will be of primary interest. An example of this type is shown in Figure 1. The stability of a binary phase can be characterized in terms of the chemical potential. If we consider a binary system whose two com- ponents are in equilibrium, thermodynamic arguments27a show that for equilibrium with reSpect to diffusion for a two component system (egg) = (g—fif) < o (7.1) Making use of the Gibbs Duhem relation, it can be shown that u - 2. ‘ u s 9. gig - x188?) and %: x165? (7.2) and so equation (7.1) is equivalent to 6:43:3(0 and gia<0 ' (7.3) For example, these conditions may be illustrated by the hexane-nitro- benzene system shown in Figure 1. If the chemical potential of hexane is plotted as a function of the mole fraction of nitrobenzene, the curves shown in Figure 5 result. Above 19°C (curve 1) a single phase only exists and the conditions of (7.3) are always satisfied. However, below 19°C, curve 3 consists of three parts: one corresponding to the phase rich in nitrobenzene, one for the phase rich in hexane, and a horizontal line joining these two, corresponding to the simultaneous presence of two phases. The curve at 19°C represents the transition temperature between these two types of curves. The horizontal portion is reduced to a single point of inflection at C which mathematically satisfies the restrictions 28 “1 2 = o .h 3,912,) (3.15:1) (7 ) and 253A (3331),, < 0 (7.5) 1 0 12., \\) Fig. 5.27a variation of chemical potential of n-hexane as a function of nitrobenzene concentration. The conditions of stability as given by equation (7.1) can be written also in terms of the free energy of mixing since we can write _ (5 F (43-191, p - ' X2 #5731)» (7.6) Therefore, for stability we can write from (7.1) and (7.6). 621: ‘b—ii) > 0 (7-7) This inequality has a simple geometric interpretation. If'F is plotted as a function of x2, then equation (7.7) requires that for a stable sys- tem to exist, the curve must be concave upward at all points. This is illustrated in Figure 6, curve 1. However, if we consider curve 2, we see that the portion of the curve BC is unstable with respect to mater- ial fluctuations and is surrounded by the portions AB and CD which are 29 materially stable but are metastable with respect to the heterogeneous mixture represented by the line AD. 'le Fig. 6.27a Variation of molar free energy with composition. The two phases in equilibrium have compositions A and D. That is, the common tangent AD satisfies the equilibrium conditions, “‘1' = “'1" (7.8) “'2' = “'2" where the primes refer to phase' and phase". FUrthermore, at the crit- ical point, we can write T -0 ___g) _ (7.9) and We will now investigate the conditions which determine whether the critical point shall be a maximum or a minimum, that is, an U.C.S.T. or a L.C.S.T. Referring to Figures 1 and 2 we have from.Prigogine and Defayz7b that the slope of the curve CA is given by a...) x2"A}' 11', + (1.1201) 11, 717— ‘ _ 7(x."-x.') xz’ or (x2" — xz') > 0 (7.11) If equation (7.10) is positive in the neighborhood of C on the curve AC, then C is an U.C.S.T., and if the slope is negative on AC, then C is a L.C.S.T. The denominator is always positive for stable phases because of (7.11) and the fact that (-g—X—:-rz) > O (7.12) Therefore, the sign of (7.10) is determined by the numerator. Now we consider two phases very close to the critical point. We let their temperature and concentrations be defined as To + 6 T, (x2)C + 6 x2' and TC +6 T, (x2)c + (5 x2". The partial molar enthalpy of com- ponent l in these solutions can be obtained from the following expres- sions a . = (ii) + (3’31) 6x . + (381) (ST (7 13) 1 1 C 8X26 2 51 C . and 8'1" = (8'1), + 7.339061." + (£1101 (7.11) The heat of transfer of component 1 from the first phase to the second is therefore M81 = H." - 81' = ($3), (51211-6..21) (7.15) From the Gibbs-Duhem relationship we can write a h x.(%,—,;)T,p + 111-3721,. = o (7.16) 31 Therefore, the numerator in (7.10) can be written —0x " (— --(§*“) 1 (0x." - 6x20 (7.17) To simplify the expression, we make use of the following relationships. The derivative of R with reSpect to x2 can be written in the form 9— : H2 _ ii, (7.18) and therefore (7.19) Since the change of component 2 in phase' must be equal to and the negative of the change of component 2 in phase", we can write 6X2' = '6x2" (7.20) Hence, if we substitute (7.19) and (7.20) into (7.17), we have for the numerator of (7.10) (6x29?- (7.21) As we have seen, the Sign of (7.10) must be the same as that of (7.21), that is, Opposite to (azH/axzz)c. Therefore, we have at an U.C.S.T. 2— (855.2% < O (7.22) and for a L.C.S.T. we have 21 (3X22) > o (7.23) These conditions relate the nature of the consolute temperature to the curvature of the line H as a function of x2. For an U.C.S.T., the curve of R(x2) must be concave up as curve 1 in Figure 7, whereas for a L.C.S.T., the curve must be convex down as curve 2. In the first case, the heat content of the mixture is larger than that of the pure 32 substances. The mixture will be formed from the pure substances with an absorption of heat, that is, it will be an endothermic reaction. In the other case, the mixture will be formed by an evolution of heat, that is, it will be an exothermic reaction. R l . t r 17,0 E10 I I I . I Fig. 7.27b Molar enthalpy as a func- tion of composition. If we consider the relation '15 = H — TS (7.21) and remember that at the critical point (azF/azxzz) is zero, we can write 2h is ( x221. = T(j5;:2)c (7.25) The curvature of R(x2) and S(x2) have the same Sign at the critical point. Therefore, in addition to (7.22) and (7.23) we have as a further condition at an U.C.S.T. CECE—2) < 0 (7.26) and at a L.C.S.T. > O (7.27) 33 It will be beneficial to express these conditions in terms of the excess thermodynamic functions. Thus we have FE = FM -|RT(x1 1n x1 + x2 1n x2) (7.28) EB = PM (7.29) 1S5 = EB - 1‘5 (7.30) The superscript E indicates the function to be the thermodynamic excess function, that is, the difference between the thermodynamic function of mixing (denoted by M) and the value corresponding to an ideal solution. So from equations (7.28, 7.29, 7.30) we can write 2—» RT 2-8 (3,5121% = $2 + (g :2)C =_-'0 (7.31) 2—E (3232),: 03—32% < O (U.C.S.T.) or > 0 (L.C.S.T.) (7.32) EBTS R ZSE 0537§2)c = _ X1X2 + (jgfiz) <50 (U.C.S.T. or > 0 (L.C.S.T.) (7.33) These conditions can be illustrated as shown below. HE X —9 1s 0 :2 1 O X —> 1 H E 2 TS Fig. 8.27C Fig. 9.2"C Thermodynamic excess functions in the neighborhood of an U.C.S.T. (Fig. 8) and a L.C.S.T. (Fig. 9). Thus for an U.C.S.T., the excess functions appear as in Figure 8 and for a L.C.S.T., they appear as in Figure 9. So it can easily be seen that the following conditions hold: for an U.C.S.T., FE:> 0 and for an L.C.S.T., >0, S >0 (7.31) 3 FE>0, EE<0, SE<0 (7.35) The conditions above can be summarized28 by saying that U.C.S.T.‘s are related to large positive deviations of the enthalpy of the system from ideality while L.C.S.T.‘S result from sufficientlylarge negative deviations of the entropy from ideality. At an U.C.S.T., the critical value of FE can arise only from ener- getic factors which affect the enthalpy of the system though not neces- sarily the entropy. However, a L.C.S.T. point will occur only if the system has a large negative excess entropy and a small negative excess enthalpy. The difficulty of satisfying the conditions for a L.C.S.T. can be shown in the following wayza. Consider a one phase system which satisfies the conditions of (7.35) and is therefore potentially capable of having a L.C.S.T. Suppose that at a temperature To the system has at O a value of PE (Point O in Figure 10). Since SE is negative,'FE will increase with T. If C: is positive, however, S8 will become less negative as T incneases and so the slope of FE(T) will decrease with increase of T. The curvature of FE(T) will be larger, the larger the ratio SE/Cg. Thus system A will show a L.C.S.T.; system B will show a closed loop and C will not show a phase separation. However, almost any system.at 0 having excess entropy greater than -FE(critical) will separate into two phases on lowering the temperature, for example, OE, OF (correSponding to a regular solution), and 00. So for a L.C.S.T. there must exist the proper combination of HE, SE, and‘CE ) but almost any system having SB larger than 4FE will show a U.C.S.T. For example, 35 E, F, G are systems showing an U.C.S.T. Since HE must change from nega- tive at a L.C.S.T. to positive at an U.C.S.T., it follows that for a closed 100p C: must be positive. FE (critical) _ Fig. 10.28 Variation of‘FE starting from 0 to L.C.S.T. (to the right) and to U.C.S.T. (to the left). An alkane polymer that is above its melting point is generally miscible in all proportions with a paraffin solvent8 and according to the original Hildebrand-Scatchard solubility parameter theory, the heat of mixing should be small and positive. The polymer is often incom- pletely miscible with aromatic and polar solvents in which the heat of mixing is large and positive. The polymer becomes more soluble as the temperature is raised until complete miscibility is attained at the U.C.S.T. However, Freeman and Rowlinson7 have shown recently that hydrocarbon polymers can also be precipitated from hydrocarbon sol- vents whether aliphatic or aromatic by raising the temperature suffic- iently above the normal boiling point of the solvent. The minimum temperature at which immiscibility occurs is the L.C.S.T. for the system. .As we have seen, in order to satisfy the thermodynamic re- quirements for a L.C.S.T., the heat of mixing must be negative at this point. 36 A modified Hildebrand-Scatchard solubility parameter theory was used by Delmas, Patterson and Somcynsky9 to Show the variation of AHM with solubility parameters 6, and 6 2. Thus AHM = 2VM/U'1/U'2(61 ‘ 6211136 1/3T) (7.36) where = vap 1/2 61 (AB, Ni) (7.37) AEivap is the energy of vaporation, Vi the molar volume, and AF, is the volume fraction of species i. As the temperature increases, AElvap will decrease. Therefore, 61 will also decrease. Since (52 is less temperature dependent than 51, a point will eventually be reached where (5 2 will be larger than (S, and the difference ((51 - (52) becomes nega- tive. So AHM will be positive or negative according to whether (5 1 is greater or less than (S 2. A quantative treatment of this problem has been given by Delmas, Patterson, and Somcynsky9 and by Bellemans and Naar-Colin29. The deri- vation of AHM is carried out using the quasicrystalline lattice or cell theory deve10ped by Prigogine3°. Thus it was shown that AHM is given by AHM no. base moles XV, = A - B(T/r,>2 (7.38) where A and B are constants which depend on the solvent-polymer system and are given by A zw NA = 267162 NA/8 (7.39) B .-)(- 10.5 (112/2512) N11 (7.10) €21, is the minimum potential energy of interaction of two segments of type i and j (a solvent and polymer segment for example). (5 is given by ‘3? * 0= (ial-Eu) (7.11) 11 37 Prigogine has shown31 that rA should be given by r, = (n + l)/2 (7.12) where n is the number of carbon atoms in the solvent molecule. Further, it was shown that RXl = AMA/T) + (BT/rA) (7.13) Thus we have an eXpression which shows the dependence of )(1 on the temperature. Or conversely, for every }(l, we can find twg correspond- ing critical temperatures. If A does not equal zero, we find the change of )(rwith T for constant rA is given by a X..- 1 _._ _ .11 .13. 31 mg. + 11R (7.11) For )(1 to be a minimum we find that 102 = (A/B)rA2 (7.15) The significance of )(1 being a minimum at this point means that above or below To, )(l'will increase and the solvent quality becomes poorer. As T increases or decreases, )(1 will reach the critical value and the polymer will precipitate from the solution. So if we solve for TC/rA, we find + 1/2 TC/rA = RX, 4031?} - MB] (7.16) Depending on the relative values of A and B, we can distinguish three possible cases. If A is zero (the polymer liquids differ only in chain length), the U.C.S.T. is zero and the L.C.S.T. is given by TC/rA = R/2B (7.17) If 1AB < (1/2 R)2, two real roots exist corresponding to the L.C.S.T. and the U.C.S.T. If 1AB > (1/2 R)2, there are no critical temperatures 38 and the polymer and the solvent are not soluble in all proportions. These three cases are illustrated in Figure ll, 12, and 13. Two TWO Phases Phases T T One Phase Two One Phase Phases X2 -—> . X2 —> Fig. 11. A = 0. Fig. 12. 1AB < (1/2 R)2, One Phase T Fig. 13. 1113 > (1/2 R)? EXPERIMENTAL Fractionation The sample of polyoctene-l used for this study was supplied by the Union Carbide Corporation. The polymer as received was a dark yellowish, rubber-like substance. No measurements were made on the bulk polymer prior to purification. The polymer was fractionated by precipitation according to the method described by Flory12d using cyclohexane as the solvent and ace- tone as the non-solvent. The solution from which the polymer was pre- cipitated was prepared by dissolving approximately ten gm. of polymer in 1000 ml. of cyclohexane. Acetone was slowly added to the stirred solution until a slight turbidity (due to precipitated polymer) per- sisted. The precipitated polymer was redissolved by warming and the resulting solution was placed in a thermostat maintained at 25i0.5°C for 21 hours to allow the precipitated phase to settle. After the two phases had separated, the supernatant phase was removed leaving the polymer rich phase in the flask. The polymer was removed from the flask by dissolving it in cyclohexane. The solution was filtered through glass wool to remove impurities and the polymer was recovered by freeze- drying. The procedure was repeated using the supernatant phase until a total of twelve fractions was- recovered. After the first fractionation, it was discovered that the discol- oration was due to a residue, probably catalyst, left in the polymer. When the polymer was dissolved in n-nonane, the residue remained insol- uble and could be removed easily by centrifugation. After the residue 39 10 was separated, the recovered polymer was tranSparent and colorless. The insoluble residue did not ignite when heated over a gas flame and an X- ray powder photograph produced a typical crystalline pattern. Thus it was concluded that the residue was predominantly inorganic in nature. In view of the amount of work entailed in extracting the residue from each individual fraction and in repeating several measurements which had been made using the impure polymer, a sample of the polymer from which the residue had been removed was refractionated in order to ob- tain pure fractions. Ten gm. of the polymer were dissolved in 1000 ml. n-nonane and the resulting mixture was centrifuged on the high-speed Servall centrifuge. The purified polymer was recovered by pouring the solution into an ex- cess of acetone. The precipitated polymer was dried under vacuum, weighed, and then fractionated according to the procedure discussed previously. This second fractionation yielded eleven fractions. In addition, several of the first fractions were purified and later used for measurements. Henceeforth, the fractions from the first fraction- ation will be designated by a primed number, and those of the second fractionation will be designated by a number followed by the letter A. Phase Studies Precipitation temperatures, Tp, of polyoctene—l in ethyl phenyl ether were determined by the method described by Shultz and Floby32. A polymer solution was prepared by weighing solvent and polymer direct- ly into a test tube. The solution in the tube was then diluted after each determination of T? to obtain the next solution. Tp was determined 11 for four to seven concentrations for each of the six polymer fractions studied. After the solution was prepared, it was placed in a stirred water bath and cooled rapidly to determine the approximate Tp. The precipi- tated polymer was redissolved by heating and replaced in the water bath. The bath was cooled Slowly (about 0.2°/min.) until the polymer precipi- tated. The temperature at which the solution first became turbid was noted, and when the solution became Opaque, the temperature was again noted. The mixture was defined as opaque when a black line behind the tube was no longer visible through the tube. The two temperatures were in most cases within 0.2°C of each other and Tp was taken to be the temperature between the two. After the polymers had been precipitated by cooling, the mixture was warmed slowly. The polymer would, in most cases, redissolve about 1°C higher than the precipitation temperature. This difference in temperature probably resulted because the warming took place too rapid- ly for equilibrium conditions to be established and because some of the precipitated phase settled. The Specific volume of the polymer between 20°C and 50°C was determined by measuring its density at 5° intervals from 20° to 50°. A cyclohexane solutipn.of the polymer was placed in a calibrated speci- fic gravity bottle and the solvent was evaporated leaving a polymer film. The bottle was filled with water and the total weight of the bottle, water, and polymer was measured over the temperature range. Thus the volume of the polymer at each of the seven temperatures was easily calculated. From a knowledge of the weight and volume of the 12 polymer, its density, and thus its specific volume, was calculated over the 30°C temperature range. The density of the solvent was calculated using the equation given in the International Critical Tables33. Viscosity All viscosity measurements were made with a No. 75 Cannon-Ubbelohde semi-micro dilution type viscometer. The intrinsic viscosities of the polyoctene-l fractions were determined in cyclohexane, bromobenzene, and ethyl phenyl ether. The temperature of the solutions was maintained to within i0.5°C by placing the viscometer in a thermostat. The solution flow times were measured by an electric timer accurate to i0.l sec. The relative viscosities of all solutions were between 1.2 and 2.0. The viscosities of three solutions at different concentrations were measured for each fraction. The first solution was prepared by weighing polymer and solvent directly into a flask whereas the other two were obtained by successive dilutions of the solution within the viscometer. Viscosity measurements in the theta solvent presented problems not encountered ordinarily when working with solutions of thermodynamically good solvents since incipient precipitation of the polymer occurs a few degrees below the measurement temperature. It was therefore essential the polymer solution did not cool below the theta temperature. All apparatus used in the transfer of the solution from the preparation flask to the viscometer was preheated to prevent precipitation of the polymer. In addition, the solvent used for the dilutions was heated above the theta temperature before it was added to the solution in the viscometer. 13 It has been shown by Cannon, Manning, and Bell that the kinetic energy correction is negligible for the type of viscometer used in this study34. Therefore, no corrections due to kinetic energy effects were applied to the viscosities. The intrinsic viscosities of all the fractions measured in bromo- benzene were equal to or less than about 1 deciliters/grm. Flory and others have shownlzes35 that for intrinsic viscosities less than about 1 deciliters/gm. it is not necessary to correct the intrinsic viscosity for rate of shear effects. The effect of shear rate on the viscosities measured in cyclohexane was investigated by plotting [7?] bromobenzene against [77] cyclohexane since some [7?] values in the latter solvent are larger than 1. As seen in Figure 11, the relationship was linear over the range of the fractions studied. This linearity indicates that no shear corrections were necessary for the visCosity measurements in cyclohexane. In addition the linear relationship between log [7?] tyclohexane and log Mw indicated that shear rate corrections were un- necessary. Osmometry The number average molecular weight of the polymer fractions was determined by osmometry. A modified Zimm-Myerson osmometer designed by Stabin and Immergut was used for the measurementsss. The instrument was designed to facilitate removal of the solution while the instrument remained in the thermostat making successive readings much faster. Another feature of the instrument is the large ratio of membrane area to capillary radius which decreases the time required to reach equilib- rium. 11 .wconconoaouo ca 2.: moH pmcmmmm ocmxmfioflozo 5 2.: moH Ho 03m oemxmfioaoho H~B m3. F. O. m. J m... N. H.. a: are a _ _ _ A a _ euezueqomoaq [a] 607 15 Since the osmotic pressure of a solution is directly pr0portional to the absolute temperature, the temperature needs to be measured to within i0.l°C. The error of the temperature measurement will not affect the molecular weight significantly. However, more stringent require- ments upon temperature control are imposed by the following considera- tions. The osmometer encloses a volume of liquid which is large com- pared to the volume of liquid per millimeter of capillary. The osmometer acts toward temperature fluctuations as if it were a sensitive thermo- meter. For example, cyclohexane at 30°C has a temperature coefficient of volume of 0.0011 ml. per ml. per °C whereas the volume of liquid contained by a 0.1 mm. length of 0.5 mm. diameter capillary is 7.8 x 10'5 m1. Therefore for an osmometer containing 10 ml. of solution, a change of 0.05°C correSponds to a change in scale reading of 0.7 mm. In order to fulfill the requirement for accurate temperature control, the osmom— etry measurements were carried out in a 30° thermostat whose temperature was controlled to within i0.0l°C. The temperature fluctuation was de- termined by a Beckmann thermometer. The membranes used in the osmometer were de-nitrated gell cello- phane grade 150. The membranes were treated in a 5% NaOH solution for one hour to render them more permeable to the solvent. After the mem- branes were treated with the NaOH solution, they were conditioned grad- ually to the solvent, cyclohexane, following the method described by Yanko37. When the osmometer was assembled, the permeability of the mem- branes was measured. The importance of the permeability measurement is to establish that the membranes are free from imperfections and that the osmometer is assembled correctly. It is also used to decide the minimum time required for the establishment of osmotic equilibrium. 16 The measurement of the time constant of an osmometer, from which the permeability is obtained, may be made with the osmometer filled with pure solvent. The initial level in the measuring capillary, hl, is set 1—5 cm. above that for the reference capillary, hz, and the descent noted at a series of times. The 10garithm of the hydrostatic head (the dif- ference between h, and hz) is then plotted vs. time. The time neces- sary for the natural logarithm to decrease by l is the time constant33. An example of this measurement is shown in Figure 15. When the results of the measurement indicated that the osmometer was functioning prOperly, the solvent was removed from the cell and it was filled with the polymer solution by means of a syringe and a long hypodermic needle. (To prevent the membranes from drying, the osmometer was never left free of solvent.) The cell was rinsed with a few ml. of solution before it was filled so that the final concentration of the solution would not be affected by the small amount of liquid which re- mained in the cell. Care was taken to remove all air bubbles from the osmometer cell after it was filled. This presented no problem since the bubbles could be seen easily through the glass cell and were easily removed. The polymer solution was preheated to the operating temperature before it was placed in the cell. However, it cooled slightly when transferred to the cell. Thus when it was placed in the cell, it warm- ed and expanded. The expansion of the solution caused an immediate capillary rise which continued for several minutes. After this expan- sion ceased, it was assumed that temperature equilibrium was established and the readings were begun. .A c< mo poflm .on pcmpmcoo 08H» ocHEempop 0» mafia w> ad moH mo poam .mH .mwm .un mafia OH m m N o m a m m H a a W .cms mom cm pampmcoo meme mHm>noucm ucmpmcoo mam» on. Y o.m 1| o.oH.rI. .50 £4 UV 501 18 Osmotic pressure was determined by measuring the difference between the level in the solution capillary, hl, (previously referred to as the measuring capillary) and the level in the reference capillary, hz. The heights, h, and hz, were measured by a cathetometer accurate to $0.001 cm. The difference between h, and h2) i.e. Ah, was therefore accurate 3 to $0.002 cm. It is usual for Ah to pass through a maximum and then decrease slowly with time. The level in the measuring capillary and therefore Ah should increase with time until an equilibrium height is reached. If Ah is plotted against time, it will increase to a maximum and then drop Slowly. The decrease in Ah (or in the level of the measuring cap- illary) is caused by polymer molecules diffusing through the membranes. Actually the polymer molecules diffuse through the membranes from the beginning of the measurements and so the maximum Ah is not the true equilibrium value. Assuming that the polymer diffusion has been constant throughout the determination, the effect can be corrected by extrapolating Ah to zero time. The intercept at zero time gives the equilibrium Ah value. An example of this extrapolation is shown in Figure 16. After the reading of the first solution was completed, it was re- moved from the cell and the cell was rinsed with pure solvent. The first solution was diluted and the procedure just described was repeated. Finally this second solution was removed and diluted and a third read- ing was made. Thus three concentrations of each fraction were measured. Ah cm 30m 2.90 2.80 2.70 2.60 19 :~ ~~~~~ ~~~~~ ~A " <3 “1Er—* ___ ,_ o F P I I I I I l I q I 1.0 2.0 3.0 1.0 , 5.0 6.0 7.0 8.0 Time Hr. Fig. 16. Osmotic equilibration curve at 30°C for polyoctene-l, F10A in cyclo- hexane. C = 0.8017 gm/lOO cc. 50 Light Scattering The weight average molecular weights of six polymer fractions were determined by light scattering. All of the measurements were performed using the Brice-Phoenix Light Scattering Photometer series 1000. Light scattering measurements were performed on six of the higher molecular weight fractions with bromobenzene as the solvent. Prelimin- ary investigation of the solvent indicated that the solvent exhibited no fluorescence at the wave length used for the light scattering measure- ments. All reported measurements were taken with the unpolarized green (Sbléfl) line of mercury. The techniques and procedures for making the light scattering measurements were, with a few exceptions, the same as those described by Mchy3Qa. One difference in technique involved the method of measur- ing the scattering intensities over the range of angles. Since the light scattering calculations depend upon the ratio of the scattering intensity at a given angle 9, Ge, to the intensity at zero angle, Go, it is impor- tant that Go remain constant throughout the measurements. However, it was found that Go would drift slightly during the time of the measure- ment and thus the ratio of 69/00 was affected. This change in the Go reading which was attributed to instrument drift or photo tube fatigue changed the 66/00 ratio by as much as two to three percent. The techni- que used by McCoy did not take into consideration the change in Go with time and, in fact, assumed it to be constant over the range of angles covered. In order to correct for this drift, it was deemed advisable to check Go after the intensity at each angle was measured. The photo tube was set at the sero angle position and the galvanometer deflection 51 was arbitrarily set at 95. The photo tube was then rotated to the first angle, the hSO position, and the intensity measured. The photo tube was returned to the zero angle position and if necessary the galvanometer deflection adjusted to 95. The intensity at the second angle was then read. This procedure was continued throughout the entire range of angles. Thus the zero angle intensity, Go, was the same for all angles. The anomalous readings between 70° and 80° mentioned by McCoy”b were also observed during this investigation. As McCoy noted, these anomalous readings were probably due to extraneous reflections. This effect was corrected by using a rectangular tube between the cell table diaphragm and the cell. This tube, although allowing the parallel light beam from the lamp to pass unhindered, fit tightly against the cell so that no extraneous light reached the photo tube. The data obtained from these measurements were treated in the man- ner outlined by McCoy with the exception that, because of larger dis- symmetry ratios, the Fresnel corrections for the back reflection of light at the glass/air interface were applied. The Specific refractive index increment, (dn/dc), of the solution due to the polymer was measured using the Brice-Halwer4° type differen- tial refractometer. The instrument was calibrated with standard sucrose and alkali chloride solutions. The (dn/dc) was measured for the whole polymer and for several fractions at the temperature at which the light scattering measurements were made. Details of the measurements and calibration procedure are given by Mchy39C. n "' If the technique just described was used, the ratio 69/00 should be independent of any instrument fluctuations. That is, G , the inten- sity of scattered light at G, is directly proportional to the incident S2 intensity, Go. Any fluctuation in Go will cause a fluctuation in 69. The ratio 66/00 will remain constant. However, it was found that the ratio 69/00 tended to decrease with time. A definite correlation was seen to exist between the decrease in this ratio and the increase in the solution temperature. The temperature of the solution increased as a result of the increase of the temperature inside the instrument. It is believed that as the temperature of the solvent increased, its solvent qualities changed which caused the polymer coil size to change. Although this effect would not change the measured value of the molecu- lar weight, it would, however, change those parameters which depend on the solvent-polymer interaction such as the second virial coefficient. Although no extensive study was made of this effect, it was found that the change did not effect the value of the data to an appreciable ex- tent and so no direct attempt was made to maintain the solution at a constant temperature. That is, the temperature of the solution was determined by the temperature inside the photometer. Lower Critical Solution Temperature Lower critical solution temperatures were made of n-pentane solu- tions of polyoctene-l and two other a-olefins, isotactic and atactic polypropylene and polybutene. For the polyoctene-l and the atactic polypropylene and polybutene, n-pentane solutions of known concentra- tions were prepared and transferred to capillary tubes by means of a long hypodermic needle. Because they were insoluble at room tempera- ture in n-pentane, the isotactic polymers were weighed directly into the tubes. The solvent also was weighed directly into the tubes. When 53 the solutions, or solvent and polymer, had been placed in the tubes, they were cooled to dry ice temperature and evacuated. The tubes were then sealed while under vacuum. On several occasions during the sealing, it was observed that solu- tion was lost due to "bumping" of the solutions while under vacuum. It was found that this problem could be solved if the solution was frozen with liquid nitrogen before the tube was evacuated. To determine if any solution was lost during the sealing, the liquid level in the tube was measured with a cathetometer before and after sealing. The tubes were heated by immersing them completely in a heating bath of Dow-Corning 550 silicone oil. The temperature of the phase separation, Tb, was found by heating the bath slowly (0.l°C per min.) until the solution became turbid. The phase separations, in most cases, took place over a range of one degree or less and were reproducible to within one degree with two or three repeated beatings. No evidence for decomposition of the polymer upon heating was observed. Above Tp, the two phases separated, the polymer rich phase being the denser. If the bath was cooled immediately after the solution be- came turbid so that the phases did not separate, the polymer would re- dissolve immediately at the precipitation temperature. RESULTS Fractionation Two samples of polyoctene-l were fractionated from a dilute cyclo- hexane solution. The first fractionation was carried out on the untreat- ed whole polymer, while the second fractionation was carried out using a sample from which the metallic residue had been extracted. This~ second fractionation yielded eleven fractions. With the exception of two extracted fractions, no studies were made of the fractions from the first fractionation. The results of the fractionation were determined by viscosity measure- ments. Thus the intrinsic viscosity of each of the eleven fractions from the second fractionation and the bulk polymer were determined in cyclohexane at 30°C. The results of the second fractionation are shown in Table I. It is seen that the molecular weights (as determined by intrinsic viscosity) of the first three fractions are in reverse order. It is suSpected that this resulted either because of non-uniform temperature throughout the fractionation or because the polymer separated on the basis of tacticity rather than molecular weight alone. Another possibility is that some branching was present in the molecules and that this caused the solubil- ity of lower molecular weight Species to be decreased. No information was available concerning either the tacticity or the possible presence of branching of the polymer and no measurements were made to determine either tacticity or branching of the fractions. 5h 55 Table I. Fractionation data. Fraction [7?] cyclo- wt. gm. wt. % [7?]iwi hexane 300 Second Fractionation 1A 6.37 0.63b 7.50 0.1.8 2A 6.72 0.133 1.5h 0.10 3A 9.30 0.290 3.h5 0.32 14A 9.6b 0.285 3.33 0.32 5A 8.60 0.7hl 8.10 0.76 6A 5.71 1.59 18.92 1.08 7A b.19 1.05 12.50 0.53 8A 2.90 0.91; 11.-19 ' 0.32 9A 1.75 1.30 15.L7 0.27 10A 1.00 0.75 8.93 0.09 11A , 0.30 0.70 8.33 0.02 Zn, 8.10 ZU’IIiwi b.29 ‘.First Fractionation 2' 6.61; 7' 2.61 8' 1.58 Bulk intrinsic viscosity, [7213 = b.57 2177],»:i = L29 56 0f the 8.52 gm. of polymer which were fractionated, a total of 8.h0 gm. were recovered for a recovery of over 98%. The fractionation data was checked by the equation [7]]8 = $;1[7?Jiwi where [7?]B and [7?]i are the intrinsic viscosities of the bulk polymer and the ith fraction respectively and wi the weight fraction of the respective polymer fractions. The value of b.29 for 3:1[77liwi is somewhat lower than the intrinsic viscosity of the bulk p6lymer, [7?] = b.57. Since the recovery of the polymer was high, this discrepancy can not be ex- plained bythe loss of higher molecular weight species. It is believed that a small amount of degradation occurred between the time the poly- mer was fractionated and the intrinsic viscosities were determined. However, this degradation was shown to be negligible over a long time period by redetermining the intrinsic viscosities of fractions. An- other possible explanation for this discrepancy is that no corrections were made for the shear effect at high intrinsic viscosity. The shear effect would cause the measured viscosity to be lower than the true value for higher molecular weight species while having no effect on the viscosity of lower molecular weight species. The overall effect would therefore cause the sum 3:1[7211wi to be smaller than [7?]B. The molecular weight-distribution of the polymer was not constructed because not enough fractions were recovered from the whole polymer for an accurate representation. Phase Studies Liquid-liquid phase diagrams were constructed for five fractions of polyoctene-l in phenetole near the critical miscibility temperature. S7 The diagrams were constructed by plotting the precipitation temperatures, Tp, against the volume fractions,/Dr2, of the polymer. The resulting phase diagrams are shown in Figure 17. The critical miscibility temperatures, TC, correspond to the maxi- mum point in the phase diagrams. In Figure 18, the reciprocal of the critical temperatures is plotted against the molecular size function, l/xl/2 + 1/2 x, in accordance with equation (3.3). The intercept of the y-axis gives the reciprocal of the theta temperature and the ratio of the intercept to the lepe of the line gives the value of the entropy of dilution parameter )(1. These values are then used with equations (1.15), (1.16), and (1.18) to calculate the excess entropy and heat of dilution respectively. These results are shown below in Table II. Table II. Thermodynamic parameters from phase equilibrh.studies. Polyoctene—l in phenetole. em) lav“?- 15,1er 1%4 cal./mgle cal./moie deg. 50.h .813 788 1.66 The value of x in equation (3.3) was calculated from.fi§ rather than fih~since it was only possible to measure Hg for two fractions. To deter- mine if the use of Mw rather than Mh to calculate x would affect the value of 1/0, a value of 1.2 for the ratio Mw/Mn was assumed and calcu- lated values of Mn were used to calculate new values of x. The important parameter 0, which is obtained at the y intercept, was unaffected by using Mw rather than Mn; however, a small change of slope is noted which introduces approximately six percent uncertainty in WI. r— 147-8 - O F2A 147.6 0 O 86.8 — O 166 -— F7A no.1; — , T °c 85.8 — 0 15.6 L- F8A 85.1; t- b5.2 — , hb.6 —- F911 hb.h #- 82.b - zee _ xof 0 ‘6\ FB' 1 I L i ' I I 242°C 1.0 2.0 3.0 8.0 5.0 6.0 {U} x 102 Fig. 17. Binary phase diagrams for polyoctene-l in phenyl ethyl ether. 59 .< 02 80pm 033338 .0 3: son.“ @3350ku .mcogomum HnoCopoohHoQ pow 09>” .mH .mE . No... x Ax N\H +m~\~X\C ma m.” NH OH EH 3.. 9.. NH .2 OH m N. w m a m N H O 1 . . . A . _ u . _ . _ . A . . . . . mo.m I 0.7m I .35 nOH X I w~.m .l mfim L Jaim i mH.m 1 07m a} 60 Light Scattering The weight average molecular weight of the polymer fractions was determined using the Zimm method as outlined previously. The Zimm plots for the polyoctene-l fractions five through ten are shown in Figures 19 through 2h. The dimensions of the polymer molecules were calculated both by the Zimm technique and Debye's dissymmetry method. To calculate the dimensions by the Zimm method, the limiting slope of the zero concentra— tion line of the Zimm plot was divided by the intercept and this ratio .set equal to (8w2/912)FZ. This value for the ratio is used because we assumed that the molecules in solution are in theform of random coils (see equation (6.h)). The wave length of the light in the solution is given by X and F2 is the mean-square end-to—end length of the polymer molecule. Since ("52)1/2 obtained by this method is a z-average length, it was converted to the weight average length by the use of equation (6.9) r2w = F2 (h + l)/(h + 2) The h is a parameter characterizing the molecular weight distribution and can be found from equation (6.8) Mn sMw (h/h + 1) Since Mn was determined from only two fractions, it was necessary to assume that the average value calculated from the two Mfi/Mh ratios was the same for all of the fractions. The average for the ratio was 1.09 and the value of h calculated from Ulis ratio was 12.10. Using this value of h, the ratio (h + l)/(h + 2) was found equal to 0.93. Therefore, r2z was multiplied by 0.93 to obtain 52w. 61 .Uomm pm ocomconoEoun cw H00 00m uoHa hpmwoomw> E30 o\mH0E\om0 0.0 0.0 .3 0.0 0.0 0. .00 .0: H \l C fiI 1. H 4.4 0.: 0.: mfg” 0.m .4.m 0.m 0.m mfim 0.0 800" H H H H H 0 _ 0om.0m 850 00800 150 00mm ®C®NCBOEOHQ 000m 000x000Hozo nu m1. I. 14 b b C 8 N. _ .H 1H 1\ N.H 0.0 0.0 m\00 O.N N.N 0.0 83 measured by light scattering and calculated from viscosity measurements using the Flory—Fox equation and I = 2.1 x 1021, are about 3.3 times as large, and the dimensions of the molecules in cyclohexane are about h.2 times as large as the freely rotating dimensions. A comparison of these dimensions are shown in Figure 30. The dimensions calculated from the viscosities measured in bromobenzene compare very well with those mea- sured by light scattering. Again the value of 2.1 x 1021 for E is shown to be correct, at least within experimental error. Finally it will be of interest to compare the dimensions of poly- octene—l with several other poly a-olefins. It has been reported by Chinai and collaborators5 that for a series of methacrylates, the ratio of the square of a linear dimension to the degree of the polymerization falls in the order n-hexyl > n-octyl > methyl > ethyl > n-butyl. The dimensions of polyoctene—l were therfore compared with several other poly a—olefins, the purpose being to discover if any such curious order existed for the poly a-olefins. A comparison was made using the ratio of (F20)1/§ to the square root of the degree of polymerization to determine if any order existed. Table IX shows the results of this comparison. These calculations indicate that the dimensions are dependent on the pendant group and are proportional to their size. Thus, with respect to the pendant group, the average end-to-end dimensions fall in the or- der phenyl > hexyl > ethyl > two methyls Qfimethyl. This order results when the data of Wilkins48 or Natta“7 are used to determine the ratio (on/D.P.)t/Z for polybutene. The ratio calculated from the data of Krigbaum45, however, results in a ratio for polybutene which is larger than both polystyrene and polyoctene-l. 8h .< Auco>aom mpocuv .85» Togo Tasman . DQ380300 ..O 0.53% «0 SEN «mcwumppmom .203 omm «ocmncwnosoua m OoOm 09038000098 000 meowmcoamc .0232» 030630000 93 mo c0325.“ .m mm 78309200 .wo 900080800006 uconopuccm muamucmoEuuoom .0m .mE nIOH X N S 0.0 m0 0.0 \0 m. _ _ 0 0 compmuom 8.0m “ 4 1 8m hcv O 4m. 0v 1 .0 320.005.0253 I l 0000 ..EmZom So 9 . m. . 0. _- oconcmn0800m V. ‘v _m .. 000w t ~\0A~..mv 338000003 N 0 400mm 85 Table IX. Comparison of (rzo/D;P.)1/2 for atactic poly a-olefins. Polymer (FZO/M)1/éx101° (1:20/'ID.P.)1/2 Pendant group polystyrene 7045 7.16 phenyl polyoctene 68 7.08 hexyl polybutene lOO46 7.h8 \ polybutene 8147 6.00 > ethyl polybutene 78046 5.81 J polyprOpylene 83.52 5.32 methyl polypropylene 9249 5.98 polyisobutylene 7650 5.66 two methyls The results of the L.C.S.T. studies substantiate the findings of other studies with bulk polymers.“8 which indicate that the L.C.S.T. decreases as the chain length of the solute species increases. Using the theory developed by Delmas, Patterson, and Somcynsky, an attempt will be made to relate L.C.S.T. to the Flory interaction parameter, IXQ. Before studying the results of the present work, it will be of interest to use the theory and the resulting equation R . + R 2 - 1/2 TC/rA = X1 [( 31%) MB] (11.1) to try to predict the results of the work carried out by Baker gt_al§ in lwhich they determined the L.C.S.T. for four samples of polyisobutylene of different molecular weights in n-pentane. The value of A used in the equation was that determined by Delmas, Patterson, and Somcysky from the heat of mixing of polyisobutylene with 86 n-paraffins. B was also that determined by the above workers; it being the value which gave the best fit for the data. The values of these constants are A 10.1 cal./mole (11.2) B 8.2 x 10'3 ca1./c1eg.2 (11.3) According to equation (7.b2), rA for n-pentane is equal to three. If we assume that the L.C.S.T. occurs at the same value of ch as the U.C.S.T., we can use the simple Flory approximation relating X10 to the molecular weight. That is, )(lc = 1/2 + (1/kl/é + 1/2 x) (11.8) Using the above values for A and B and the Flory approximation for )(10, the TCL values for the polyisobutylene in n-pentane were calculated. The results of the calculations are shown in Table X. The data shown is that of Freeman and Rowlinson7 and Baker $3.31'8 Table X. The thermodynamic interaction parameter as a function of molecular weight and the observed and calculated TCL for polyisobutylene in n-pentane. Polymer M 1/x1/2 X10 TCL(°K) TCL(OK) (observed) (calculated) (ref. 7) 5 1.5810000 .007 .507 3L8 33h IIIa 111,000 .086 . 586 373 396 ms 62,000 .0110 .510 366 360 VB 2,250,000 .007 .507 31.1. 3311 If we plot TCL against 1/k1/2, we obtain the relationship shown in Figure 31. Although this plot is not strictly justified by equation (11.1), there should be'a linear dependence of TCL or 1/k1/2 if the factor hAB is 87 much smaller than R( X1C)2. Polymer II from Baker _e_t _a_l_. does not fit this linear relationship. It is believed that the molecular weight of polymer II was too small (1,170) to justify using the preceding method for calculating T Also the fact that unfractionated polymers were CL' used would lead to uncertainty in determining the precipitation temper- ature. That is, TCL values measured for polymers having true weight average molecular weights equal to the molecular weights given in Table X would probably be different than those observed. An attempt was made to calculate the A and B from the observed data of Freeman and Rowlinson. Two equations were obtained by substituting two sets of the data, i.e., values of )(lc and T into equation CL’ (11.1) and the equations were solved for A and B. It was found that the values of A and B are very sensitive to T and unless accurately C known values are available, these constants can not be obtained by this method. It should be mentioned that the )(IC calculated from thermodynamic data by Baker 33 al. for polymer V is somewhat larger than the )(lc calculated from.the Flory approximation. However, the value for polymer V of )(lc = .53 givesTCL = 3b6°K which is much closer to the Observed value of 3hh0K. Thus the use of the Flory approximation for calculating )(1c to determine the L.C.S.T. should be taken with reservation and at best should be considered an approximation.which can be used if thermodynamic data are not available. Independent values for A and B were not available for polyoctene-l and so it was necessary to assume A equal to zero and to calculate B .0. ..000 SUV 0000 w ..000 Aav 0C00zusnom0x0oa «Anvv 010000005000 .000000010 00 000000010 >000 030 000 ~\ \0M\H0. 000000000 000m 000500005 0:0 00000m0 00300000300 00003000 00000000 00300 0:0 00 000 0.0 00. mo. -50. 00. mo. 40. m0. «0. .0m .0000 88- _ 0 1 0 0 l 0 J 0 0 I 0% 0mm 0.0m 00h 0mm 1 0000 l 0m: 0m: 89 from one of the TC values. A is equal to zero if the solvent and poly- mer differ only in chain length. Therefore, this assumption is not un- reasonable since the polymer is predominately methylene groups. Also if A is zero, the U.C.S.T. should be zero. A solution of polyoctene-l F5 (Mw = h x 105) was cooled to the freezing point of the solvent (1h20K) without observing precipitation. Thus this criteria is obeyed, at least as far as can be observed. Hence if A is equal to zero, equa- tion (11.1) becomes = / T‘C/rA R)QC,B (11.5) If we use the TC for F5A, B is equal to 6.8h x 10-3 cal./deg.2. The results of the calculation of T for the remainder of the fractions C are shown in Table XI. Table XI. The thermodynamic interaction parameter as a function of molecular weight and the observed and calculated T for polyoctene-l in n-pentane. CL Fraction Mw x 10“6 1/01/3 ‘Xfic TCL(OK) TCL(°K) (observed) (calculated) 5A 1.32 .005 .505 1.38.1 1138 6A 1.18 I .009 .509 1139 881 8A .75 .010 .510 th th 11A .06 .OhO .580 hh6 b68 An attempt was also made with this data to calculate A and B. How- ever, as before, it was not possible to obtain satisfactory results from the TC data alone. 90 The agreement of the calculated L.C.S.T. with the observed values becomes worse as the molecular weight increases. This indicates that A is not equal to zero as was assumed and/0r that, as was decided before, the interaction parameter can not be calculated accurately by the Flory approximation. So in order to be able to calculate the L.C.S.T. with more ac- curacy, it is necessary that A and B be determined independently or from highly precise L.C.S.T. data and that true values for)(1c be known. Although the calculations do not fit the observed data exactly, the equation seems to fulfill the need of predicting the dependence of L.C.S.T. on molecular weight. . If we extrapolate the TCL to infinite molecular weight, i.e., to 1/kl/é = 0, we obtain the L.C.S.T. for infinite molecular weight poly- mer. This is an analogy to the familiar Flory theta temperature which is the U.C.S.T. for polymer of infinite molecular weight. 0n the basis of this analogy to the 0U for U.C.S.T., we will tentatively refer to the L.C.S.T. for infinite molecular weight as 0L. Using equation (11.5) and a value of 0.5 for)(u:, the TCL'S for polyisobutylene in n-pentane and polyoctene-l in n—pentane were calcu- lated. This is compared with 0L as determined from the extrapolation just described. (See Figure 31) The results of the calculation and the extrapolation are shown below in Table XII. The results for polyoctene-l agree to within less than 1%. This agreement is very good considering the assumption made. The agreement for the polyisobutylene results is about 3%. This agree- ment indicates that equation (11.5) predicts values of T fairly well CL if the proper)(1c is used. 91 Table XII. values of 0 obtained by extrapolating T to infinite molecular weight and calculating from equation 11.5 using KC = 0050 Polymer 0L 0L (extrapolation) (calculated from 11.5) polyisobutylene 3430K 3380K polyoctene-l b37°K h3h°K SUMMARY A sample of atactic polyoctene-l was fractionated and characterized by light scattering, osmometry, viscometry and phase equilibria measure- ments. Molecular weight-viscosity relationships were established for the polymer in cyclohexane, bromobenzene, and phenyl ethyl ether. The mean—square end-to-end dimension of the polymer in bromobenzene was calculated from light scattering data using both the Zimm method and Debye's dissymmetry technique. The dimensions calculated by the two methods agree within experimental accuracy. The average value of the universal hydrodynamic parameter, E, was found equal to 2.01 x 1021. This agrees very well with the most commonly accepted value, 0 = 2.1(i0.2) x 1021. This close agreement indicates that the polymer molecules in solution are in the shape of random coils (as determined by a Gaussian distribution) and so can be treated in accordance with the standard theories of dilute solutions. The molecular weights of all but two fractions were too high to be studied by osmometry. The dimensions of a series of poly a-olefins were compared by calcu- lating the ratio of the average end-to-end dimensions of the polymers. It was found that the order, with respect to the size of the pendant groups, was phenyl > hexyl >xethyl > 2 methyls 5% methyl. This is the order expected since the larger groups should cause the polymer molecules to be more expanded due to a higher probability of chain interference. The lower critical solution temperatures, L.C.S.T., for a series of solutions of polyoctene-l fractions in n-nonane were measured. The decrease in TCL with increase in molecular weight was tentatively 92 93 explained in terms of the Flory interaction parameter, )(1. .Also it was shown that fairly accurate values of T could be calculated using )(1 CL calculated from the molecular weight or thermodynamic measurements using a relationship based on the Prigogine cell model of solution. It was found that a plot of T against 1/01/2 gave a linear relationship. CL The intercept of this line at infinite molecular weight, i.e., at l/xl/2 = 0, was defined as 0 that is, the L.C.S.T. for polymer of infinite L) molecular weight. The parameter, 0L, was also found by calculating from the relationship mentioned above using )(lc = 0.5. The two values agree to within less than 1% for polyoctene-l. Thus the relationship between TCL and X1 appears to be correct for the system investigated. QUIT—“Lo \1 10. 11. 12. NHHI—‘Hl—‘Hl—J O\O CDNQU'LJZ‘K» |'\) '01 F” P‘ E: F1 7* 3’ F3 F3 7’ BIBLIOGRAPHY . Natta, Makromol. 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