1366
`
`A.facromolecules 1986, 19, 1366~1374
`
`by the temperature change of a few degrees near 7'c of the
`liquid crystal. The viscosity of the nematic phase of EBBA
`is in the same range as that of liquid water, and the liq(cid:173)
`uid-crystalline domain supposedly acts as the efficient
`mobile region of the gases. It was concluded by these
`authors that the temperature dependence of P is influ(cid:173)
`enced by the thcrn1al n1otion of the membrane component
`as well as by the continuity and/or size of the liquid-crystal
`phase.
`An analogous argument may be presented for inter(cid:173)
`pretation of the permeation characteristics of the inulti(cid:173)
`component bilayer film. Discontinuous jump of Pat Tc
`is ascribable to greater gas mobility in the liquid-crystalline
`phase of the hydrocarbon component, and the jump be·
`comes larger as the hydrocarbon domain is enlarged.
`Concluding Re111arks
`Multicomponent bilayer membranes can be immobilized
`in the form of PVA composite films. The DSC and XPS
`data indicate that the hydrocarbon and fluorocarbon bi·
`layer components are phase.separated and that the fluo·
`rocarbon component is concentrated near the film surface.
`'fhese component distributions produce favorable effects
`on pern1selectivity of 0 2 gas. The selectivity (P02 / PNJ is
`apparently detennined by the surface monolayer (or IaYers
`close to the surface) of the fluorocarbon component, and
`the penneability is pro1noted by the presence of large
`domains of t-he fluid (in the liquid·crystallinc state) hy·
`drocarbon bilayer.
`
`Acknowledgment. \Ve extend our appreciation to the
`Asahi Glass Foundation for Industrial Technology for fi.
`nnncial support-.
`Registry No. l, 91362·66·2; 2, 89373·65·9; 3, 100993-84-8; 4,
`100993·85·9; 5, 82838-66-2; 0 2, 7782-44·7; N,, 7727-37·9.
`
`References and Notes
`(1) Nakashima, N.; Ando, R.; Kunitake, T. Chem. Lett. 1983, 1577.
`(2) Shimomura, l'-.1.; Kunitake, T. Polym. J. (Tokyo) 1984, 16, 187.
`(3) Higashi, N.; Kunitake, T. Polym. J. {Tokyo) 1984, 16, 583.
`(4) Kunitake, T.; Tsuge, A.; Nakashima, N. Chem. Lett. 1984,
`1783.
`(5) J(unitake, T.; Higashi, N.; Kajiyama, T. Chem. Lett. 1984, 717.
`(6) Takahara, A.; Kajiyama, T., persoiial com1nunication.
`(7) Kunitake, T.; Tawaki, S.; Nakashima, N. Bull. Chem. Soc.
`Jpn. 1983, 56, 3235.
`(8) Kunitake, T.; Higashi, N. J. Am. Chem. Soc. 1985, 107, 692.
`(9) Kunitake, T.; Asakuma, S.; Higashi, N.; Nakashima, N, Rep.
`Asahi Glass Found. Ind. Technol. 1984, 45, 163.
`(10) Okahata, Y.; Ando, R.; Kunitake, T. Ber. Bunse11-Ges. Phys.
`Chem. 1981, 85, 789.
`(11) Barrer, R. 11.; Skinow, G. J. Polym. Sci. 1948, 3, 549.
`(12) Hoffman, S. "Depth Profiling~ in Practical Surface Analysis;
`Briggs, D.; Senk, i\1. P., Eds.; \Viley: New York, 1983.
`(13) Brundle, C.R. J. Vac. Sci. Techno/, 1974, 11, 212.
`(14) Kajiyama, T.; Nagata, Y.; \Vashizu, S.; Taknyanagi, i\f. J.
`J.fembr. Sci. 1982, 11, 39.
`(15) Kajiyama, T.; \Vashizu, S.; Takayanagi, l'-.t. J. Appl. Polym.
`Sci. 1984, 29, 3955.
`(16) \Vashizu, S.; Terada, I.; Kajiyama, T.; Takayanagi, l'-.t Polym.
`J. (Tokyo) 1984, 16, 307.
`(17) Kajiyama, T.; \Vashizu, S.; Ohmori, Y. J, Afembr. Sri, 1985, 24,
`73.
`
`Phase Equilibria in Rodlike Systems with Flexible Side Chains t
`
`M. Ballauff
`.i\fax-Planck-lnstitut far Polymer/orschung, 65 f.fainz, FRG. Receiued July 31, 1985
`
`ABSTRACT: The lattice theory of the nematic state as given by Flory is extended to systems of rigid rods
`of axial ratio x and appended side chains characterized by the product of 2, the number of side chains per
`rod, and m, the number of segments per side chain. Soft intennolecular forces are included by using the familiar
`interaction parameter. Special attention is paid to the form of the orientational distribution function, which
`is introduced in terms of the refined treatment devised by Flory and Ronca as well as in its approximative
`form given by Flory in 1956. At nearly a.thermal conditions the theory presented herein predicts a narrowing
`of the biphasic gap with increasing volume fraction of the side chains. The region where two anisotropic phase.s
`may coexist becomes very small in the presence of side chains. It is expected to vanish at a certain critical
`volume ratio of side chains and rigid core of the polymer molecule. The dense anisotropic phase at equilibrium
`with a dilute isotropic phase at strong interactions between the solute particles (wide biphasic region) is predicted
`to beoo1ne less concentrated with increa.w of the product of z and m. The deductions of the treatment presented
`herein compaze favorably with results obtained on lyotropic solutions of heliral polypeptides like poly(7·benzyl
`glutan1ate).
`
`During recent years there has been a steadily growing
`interest in thermotropic and lyotropic polymers. Spinning
`or injection molding in the liquid-crystalline state can lead
`to fibers of high strength and stiffness. 1 However, the
`1nelting point of typical aroinat.ic polyesters exhibiting a
`1nesophase often exceeds 500 °C, 2 which makes the pro·
`cessing of these n111terials by conventional methods very
`difficult. In order to reduce tho melting temperature of
`thermotropic polyn1ers, flexible spacers have been inserted
`between the mesogenic units. 3 But these semiflexible
`polymers no longer possess the rigidity necessary to pro(cid:173)
`duce the desirable mechanical properties:l Alternatively,
`sufficient solubility and lo\ver melting point may be
`
`tDedicated to the memory of Professor P. J, Flory.
`
`achieved by appending flexible side chains to rigid rod
`polymers. This third type of liquid·crystalline polymer
`has been the subject of a number of recent studies. Lenz
`and co·workers showed that substitution with linear or
`branched alkyl chains5•6 considerably lowers the melting
`point of poly(phenylene terephthalate). Similar observa·
`tions have been n1ade in the course of a study of poly(3·
`n-alkyl-4-hydroxybenzoic acids).7 Gray and co-workers
`demonstrated that side-chain·modified cellulose derlva·
`tives exhibit thern1otropic as well as lyotropic behavior.8•9
`The solution properties of poly(7-benzyl glutamates)
`(PBLG) are directly influenced by the presence of flexible
`side chains, as has been pointed out by Flory and Leo·
`nard.10 The thermodynamics of PBLG in various solvents
`has been investigated theoretically as well as experiinen·
`tally by Miller and co·workers, 11•12 leading to the conclusion
`
`0024-9297 /86/2219·1366801.50/0 © 1986 American Chemical Society
`
`Roxane Labs., Inc.
`Exhibit 1022
`Page 001
`
`

`
`Afacromolecules, Vol. 19, No, 5, 1986
`
`Rodlike Syste1ns with Flexible Side Chains 1367
`
`the lines given in ref 13. The number of situations
`available to the first segment of the first submolecule thus
`follows ns the number of vacancies n0 -fj. If there is no
`correlation between the adjacent rows parallel to the axis
`of the domain, the expectation of a vacancy required for
`the Yi+l - 1 first segments of the remaining submolecules
`is the volume fraction of the vacancies (n0 - fj)/n 0• If a
`given site for the first segment of a submolecule is vacant,
`the site following it in the same row can then be only
`occupied by either the initial segment of a subrnolecule or
`a segment of a side chain. The conditional probability
`therefore is given by the ratio of vacancies to the sum of
`vacancies, submolecules, and total number of the segments
`of the side chains (n 0 - /j)/ (n0 - fj + 2:/Yi + zmj). Inas·
`much as we assume that there is no correlation of the
`conformation of the side chains \Vith the orientational
`order (see above), the expectancy of a vacancy for the zrn
`segments of the side chains is given by the volume fraction
`of etnpty sites (n 0 - fj)/n 0• It follows that
`"i+l = (no - fj) x
`
`. ),-,,., x
`no - f; + LYi + zm;
`( 11011~ fi )'m (l)
`
`( 110:,fj y..-•(
`
`. 110 ~ fj
`
`or
`
`j
`v;+1 = (no - fj)f(no - xj + Ly,)>'in-1nol-yJu-1m
`
`(2)
`
`\Vith negligible error we may write
`
`(no - /j)!
`Vj+l = no - fU + l)!
`
`j+l
`(n0 - xU + 1) + LY;)!
`-----~---n0 t-YJ+1 -lm (3)
`j
`(n 0 - xj + LY;)!
`The results deduced by the lattice treatment are ex·
`pected to become less secure at greater disorientations, i.e.,
`at larger values of y/x. For the ordered phase under
`consideration here, this problem should be of 1ninor im(cid:173)
`portance, ho\1,•ever. As usual,m the configurational partition
`function ZM may be represented as the product of the
`combinatorial part Zu:rnb• the orientational part Zorient• and
`the intrarnolecular or conformational part Zcour:
`
`(4)
`
`Since the configuration of the side chains is assu1ned to
`be independent of the order of the respective phase (see
`above) 1 Zc<tnf may be dismissed in the subsequent treat(cid:173)
`ment. The con1binatorial part is related to vi by
`Zromb = (_t_,)ftvi
`n.,.
`
`(5)
`
`(S)
`
`and thus follows as
`_ (no - n1 x + Jnz)! n,(l-J-zm)
`no
`Zwmb -
`I
`I
`n.,.n 1•
`where n 1 = n0 - fn:r. denotes the number of solvent particles
`and n:r. the number of polymer molecules. As usual, J is
`the average value of the disorder index y defined by
`n,,
`5' = L-y
`n,
`with ltz>' being the number of molecules whose rigid core
`is characterized with regard to orientation by y. Follo\ving
`
`(7)
`
`Figure l. Representation of a rigid rod with appended side chains
`in a cubic lattice.
`
`that the presence of flexible side chains in this polymer
`profoundly alters the phase diagrams as compared to the
`rigid rod 1nod.el. In this work we present a comprehensive
`treatment of the thermodynamics of stiff rods with ap·
`pended flexible side chains. As in the theory of \Vee and
`Miller,12 it is based on the lattice model originated by
`Flory, 13·H which previously has been shown to account for
`all experimental facts obtained from suitable model sys·
`tems. 1s-19 Special attention is paid to the form of the
`orient.ational distribution function, as expressed in tern1s
`of the Floey~Ronca theoryX>.2l or in the 1956 approximative
`treatment. 13 In order to treat lyotropic systems soft in(cid:173)
`tern1olecular forces are included by means of the familiar
`interaction parameter. Steric constraints between adjacent
`side chains appended to the same rod, an important
`problem in the course of a statistical-mechanical treahnent
`of membranes22 and semicrystalline polymers,23 are not
`taken into account in the present treatment It is assumed
`that the distances betw·een neighboring side chains are
`sufficiently large, as is the case in the polyester systems
`mentioned above.5•7 However, as discussed by Flory and
`Leonard,10 this effect may come into play when helical
`polypeptides such as PBLG are considered.
`
`Theory
`The system considered here consists of rodlike n1olecules
`\vith flexible side chains appended regularly (cf. Figure 1).
`As is customary, we subdivide the lattice into cells of
`linear dimensions equal to the dian1eter of the rodlike part
`of the particle. Thus the number of segtnents comprising
`this part is identical with its axial ratio, henceforth denot.ed.
`by x. For shnplicity, we take the segments of the side
`chains and the molecules of the solvent to be equal in size
`to a segment of the rodlike part. The axial ratio of the
`solvent therefore is given by xK>h·<nt = 1. To each rod z side
`chains are appended comprising m segments. Since the
`side chains are not incorporated in the main chain, they
`are not expected to take part in the course of the ordering
`transitions. Therefore their configuration may be assumed
`to remain unaffected by the nature of the respective phase,
`Consider a phase or liquid-crystalline domain where the
`rodlike parts of the molecules are preferentially oriented
`to the axis of the domain, the latter being taken along one
`of the principal axes of the lattice. As in ref 13, the stiff
`part consists of y sequences containing x/y segments (cf.
`Figure 1). The relation of y to the angle ..;, of inclination
`to the axis of the domain \Vill be discussed later in this
`section.
`The calculation of the configurational partition function
`rests on the evaluation of the number vi+l of situations
`available to the j + 1 n)olecule after j particles are intro·
`duced. Let n0 denote the total number of lattice sites and
`f = x + niz the overall number of segments per molecule.
`Pursuant to the calculation of vi+l• it is expedient to in·
`traduce the rigid part first. Doing this we proceed along
`
`Roxane Labs., Inc.
`Exhibit 1022
`Page 002
`
`

`
`1368 Ballauff
`
`Flory and Ronca, 20 the orient.ational part of Zoofnt becomes
`
`Zorirnt = TI _!__.'
`aw II ) ' "
`(
`n,,)
`Y
`
`(8)
`
`As in ref 20, wy denotes the a priori probability for the
`interval of orientation related to y, and u is a constant.
`Introducing Stirling's approximation for the factorials, eq
`4 together \vith eq 5 and 6 gives
`u,
`-In Z,.1 = 11 1 In v1 + n,, ln -
`.
`x
`(110 - 11,X + yn,) In [ 1 - u,( 1 - ; ) ] -
`nr(l - Y - zni) + n.r~ 1l.ry In [ n.ry _!. ]- nx In <! (9)
`
`Y nx
`
`1lx Wy
`
`-
`
`where the quantities v1 = n1/n0 and Vx = n~x/11 0 denote
`the volume fractions of the solvent and of the rigid core
`of the polyiner, respectively.
`For evaluating the orientational distribution 've take the
`last n1olecule inserted into the lattice as a probe for the
`orientationnl order at equilibrium (see ref 20). Hence,
`equating j + 1 in eq 2 to nu we obtain
`
`which imn1ediatcly leads to
`1wy exp(-ay)
`nx>·/nx = / 1-
`\vith the quantity a defined by
`
`a = -In [ 1 - u,( 1 - ~) ]
`
`and
`
`!1 = Lwy exp(-ay)
`y
`
`(10)
`
`(11)
`
`(12)
`
`(13)
`
`Alternatively, eq 11 may be derived by variational
`methods. 20 Note that the orientational distribution
`function nx .fnx is given by the same expression as deduced
`in ref 20. Thus the functional dependence of nxy/nx on
`the disorientation index y of the rodlike part remains
`unaffected by the presence of flexible side chains. Ac(cid:173)
`cordingly, Zoiftnt follows as
`Z0ii~nt = nx[ln (/1cr) + ayJ
`and, upon insertion into (9)
`
`(14)
`
`-In Z = n1 In v1 + 11, In ~ - (n0 - n,x) In [ 1 -
`u,( 1 - ~)] - n,(1 - y - zm) - n, In (f1u) (15)
`
`For examining the effect of small interactions between the
`solute particles, the heat of mixing AHM is incorporated
`into the expression for the free energy. If it is assu1ned
`that randomness of mixing is preserved at all degrees of
`orientation, ARM may be taken to be proportional to the
`number of segments of one type and the volume fraction
`of the other species. Hence, introducing the fatniliar in(cid:173)
`teraction paran1eters x 12, x 13, and x23, ARM becomes
`UHM = X12n1v2 + X1sn1V1 + X2sn2Vs
`(16)
`where the index 1 refers to the solvent, 2 to the rigid part
`
`Afacromolecules, Vol. 19, No. 5, 1986
`
`of the solute particle, and 3 to the flexible side chains. The
`quantities Xii are given as usual 2~ by
`
`(17)
`
`where xi represents the number of seginents per interacting
`species, Zc is the coordination number, and Awij (=w1; ~
`0.6(wu + wi1)) denotes the change in energy for the for(cid:173)
`mation of an unlike contact. Since the volume fraction Vs
`of the side chains depends on Vz through v0 = vlzm/ x, we
`have
`
`with
`
`and
`
`zm
`X2 = ~Xn
`
`(18)
`
`(19)
`
`(20)
`
`The chemical potentials are derived from eq 7 and 18 with
`the stipulation that at orientational equilibrium
`
`In ZM )
`(llzy/nz) ni.n, = 0
`
`(
`
`Hence
`
`(21)
`
`and
`
`6µ,' /RT= In ";' - t( ui' + ";') + f - u/11 -~) +
`zma - In (/1cr) + ( 1 - vz'~ )( v1'xx1 + vz'z: x2) (22)
`In what is to follow all quantities referring to the aniso(cid:173)
`tropic phase are marked by a prime. The corresponding
`expressions for the isotropic phase, obtained by equating
`ytox,are
`6µ 1/RT =
`
`In u1 + u, + u,( 1-0 + u,[ (u, + u,)x 1 - ~ ';'x,]
`6µ,/RT =In~+ f + t( u1 +~)-In u +
`( 1 - Vx~ )( U1XX1 + v/~: X2) (24)
`
`(23)
`
`For evaluating the composition of the conjugated phases
`by means of the equilibrium conditions
`Aµ 1 :::: Aµ1'
`A112 :::: Uµ2'
`
`(25)
`
`(26)
`
`the relation between y and the angle of inclination has to
`be specified. Since the orientational distribution as rep·
`resented by eq 11 does not change when flexible side chains
`are appended to the stiff rods, we may directly use the
`expression given by the Flory-Ronca treatn1ent of the
`lattice model. 20 Thus the disorder index y is represented
`by
`
`Roxane Labs., Inc.
`Exhibit 1022
`Page 003
`
`

`
`Macromolecules, Vol. 19, No. 5, 1986
`
`Rodlike Systems with Flexible Side Chains 1369
`
`,
`4
`I
`y = -X SIO 'f
`.
`In orientational equilibriun1 its average value follows as
`
`(27)
`
`~
`
`and the order parameter s beco1nes
`s = 1-';,/,//1
`with the integrals /p being given by
`r'''
`fp = Jo
`sinP <./; exp(-a sin <./;) d<./;
`
`The factor a relates to a (cf. eq 12) by
`4
`a == -ax
`
`~
`
`(28)
`
`(29)
`
`(30)
`
`(31)
`
`For values of a >> 7r /2, the integrals fp may be developed
`in seriesf yielding
`
`!1 = l_ + 0(1..)
`f, = ~ + 0(1..)
`f, = ~ + 0(1..)
`
`a2
`
`a4
`
`o:3
`
`a5
`
`a4
`
`a6
`
`(32)
`
`(33)
`
`(34)
`
`In this asymptotic approximation20 the average disorder
`paran1eter y directly follows from
`2
`y=~=
`111 [l - v,(l - y /x)]
`a
`Thus, we obtain for the chemical potentials eq 23 and 24
`the expressions
`
`(35)
`
`(36)
`
`D.µ//RT=
`v,' {
`ln -
`-
`X
`
`v,')
`v ' + -
`X
`I
`
`( Y)
`+ f - v ' 1 - -
`X
`t)[
`In Y) - ln ;2 - C + 1 - v/~ v1'xx1 + v/-;-x2
`]
`(
`zm
`u
`
`%
`
`2zm
`+ -_- + 2(1 -
`J
`
`(37)
`
`where C = 2 In (1re/8), Further simplification can be
`achieved by omitting the small constant C and equating
`(f to x (1956 approximation; cf. ref 11). In order to obtain
`the volume fraction of the polymer in the coexisting phases,
`trial values of v:r' and a (cf. eq 12) are chosen which serve
`for the numerical evaluation of the integrals fp through eq
`30. If the quantity a (eq 31) exceeds 30, the /p may be
`calculated by the series expansion eq 32-34 using the first
`four tern1s, \Vith the fp being known, y follo\vs frotn (28),
`which in turn leads to an improved value of a. The whole
`calculation is repeated until self-consistency is reached,
`Equation 26 is then solved numerically to obtain v%, from
`which an improved value of v,/ is accessible via eq 25. The
`entire calculation is repeated with a new value of v,/ until
`co1npliance with all conditions for equilibrium is reached.
`\Vhen the 1956 approximation is used, the same sche1ne
`
`applies to eq 25 and 26, with eq 21 and 22 being replaced
`by eq 36 and 37. Here, self-consistency directly follows
`from eq 35. The volume fractions u/ and v/' of two an(cid:173)
`isotropic phases in equilibrium can be evaluated by a
`similar nun1erical solution of (25) and (26) after insertion
`of (21) and (22) together with (28), Since this calculation
`is inore sensitive toward the orientational distribution
`function, the exact treatment of Zoiient has to be used.
`
`Results and Discussion
`Parts a-<l of Figure 2 sho\ct the phase diagrams resulting
`for rigid rods of axial ratio x = 100 with flexible side chains
`characterized by the nun1ber of side chains z per rod and
`the number of segments ni per side chain.
`From the theoretical treatment it is evident that only
`'rhus the poly1ner molecules
`the product zm matters.
`under consideration here are fully determined by x and
`zm. Since the seginents of the side chains are assumed to
`be isodiametric with the segments of the rigid core, the
`quantity zni/x directly gives the ratio of the volumes oc(cid:173)
`cupied by the respective parts of the molecule. For the
`present purpose of a discussion of the influence exerted
`by the side chains, the parameter Xt (see eq 19) measuring
`the interaction of the solvent and the rigid core \vill suffice
`to take into account intermolecular forces. 'l'he second
`pBiameter x2 related to the interactions of the side chains
`with the rigid core only matters at very high concentrations
`and is thus of minor importance.
`Figure 2a corresponds to a system of rigid rods immersed
`in a solvent. The dashed lines indicate the metastable
`continuations of the phase lines. This diagram has been
`calculated by Flory in the famous work13 in 1956 using the
`approximate forn1 of the orientational distribution (see eq
`36--38). 'rhe result given herein was obtained by resort to
`the treatn1ent of the lattice model originated by Flory and
`Ronca. 20 A comparison of both n1ethods shows that the
`1nain features of the diagram are rather insensitive toward
`the respective form of the orientational distribution
`function n%y/n%. However, the stability of the equilibria
`between two nematic phases is profoundly influenced by
`niy/ni. This fact becomes even more iinportant when
`going to lower axial ratios x. For example, the Flory-Ronca
`treatment predicts a triphasic equilibrium already for x
`= 50, an axial ratio not being sufficient to produce a stable
`neinatic--nematic equilibriun1 in the frame of the 1956
`approxiination. From these results it is obvious that the
`exact treatment of nx /n% is necessary for the precise
`calculation of the triphasic equilibria but is sufficiently
`accurate for evaluating the isotropic-nematic equilibrium
`at axial ratios greater than 100.
`As can be seen fro1n a cotnparison of Figure 2, part a,
`with parts b-d, there is a marked influence of the side
`chains on the phase behavior of rigid rcxls. \Ve first discuss
`Figure 2, part a, 2a together with part b referring to zm
`= 0 and z1n = 2, respectively. Here vP and up' denote the
`total volume fractions of the polymer in the isotropic phase
`and in the coexisting ordered phase. Both phase diagrams
`exhibit the same gross features. At x 1 = 0 the soft in(cid:173)
`teractions between the rods and the solvent are null,
`corresponding to athermal conditions. The narrow bi(cid:173)
`phasic region at this point is continued until a pair of
`reentrant nematic phases appears at a critical point noted
`on the right-hand side of the ordinate. At a given value
`of x1, in the regime just beyond the critical point, either
`of two pairs of phases n,i or 11,n may occur. Further in(cid:173)
`crease of x1i equivalent to a lowering of temperature, leads
`to a triple point where three phases are in equilibriu1n.
`Raising x1 to still higher values is followed by a dilute
`isotropic phase in equilibrium with a dense nematic phase.
`
`Roxane Labs., Inc.
`Exhibit 1022
`Page 004
`
`

`
`1370 Ballauff
`
`Alacromolecules, Vol. 19, No. 5, 1986
`
`.,,_~ -·---------;1
`I
`x,,
`I"
`
`!
`
`•
`
`ti
`
`'
`
`(
`I
`.. 1
`
`' . '
`
`I
`
`"'
`co ~~o, °'
`
`I
`
`'
`
`!~
`
`\/
`
`'
`"" OILj
`I
`
`""I
`"'' ' cc;;; I
`
`001·
`
`..
`
`"
`
`"
`"
`I
`
`I
`
`02
`
`00
`00
`
`02
`
`v .. ~; . . . • .
`
`IQ
`
`08
`
`~.io '...,
`•
`
`0'
`
`---·---,1
`
`'1:0
`•
`l<T·•2
`
`06
`
`o:i:.· ". o, ·-o;--~-- ·;;---·--·-;~
`""
`---------,
`
`,,
`
`"
`"
`"
`
`"I "" ...
`
`i
`
`L 00
`
`,. '
`
`•• :c.o
`,,, .. J)
`
`"'
`
`"'
`
`Figure 2, Phase diagrams of rigid rods imn1ersed in a solvent;
`xis the axial ratio of the rigid core, z is the number of side chains
`per rigid core, and m is the number of segments per side chain.
`The solvent power is measured by the interaction parameter Xt·
`The quantities Up and u% denote the volume fraction of the tot.al
`polymer and its rigid core, respectively.
`
`Several features command attention when Figure 2, parts
`a and b, are compared. Going fron1 2m = 0 (perfectly rigid
`rods) to zm = 2 narro\VS considerably the region \vhere to
`nematic phases are coexisting. If zm > 4, this region be-
`
`• •
`02i::~~~~~-~--"'~~s~o~'-•o..,
`xdOO
`"
`
`00
`
`02
`
`06
`
`08
`
`10
`
`Figure 3. (a) Volume fraction up and l!p1 of the polyn1er in the
`coexisting i.sotropic and anisotropic phase, respectively, at athem1al
`conditions vs. the volume fraction u.' of side chains in the an·
`isotropic phase. (b) Volume fraction l.!z and u/ of the polymer
`in the coexisting isotropic and anisotropic pha.se, respectively,
`at athermal conditions vs, the volume fraction v.' of side chains
`in the antisotropic phase .
`
`comes unstable for x = 100. Note that both the critical
`value of x 1 and it.s value at the triple point are increasing
`with appending of side chains. Furthermore, the aniso(cid:173)
`tropic phase coexisting with the isotropic phase above the
`critical point is less concentrated when compared to im
`= 0. 'rhus, even a small volume fraction of side chains as
`expressed by the ratio of zm to x suffices to change the
`phase behavior profoundly. This is even 1nore obvious
`when higher values of zm (Figure 2c,d) are used. Here,
`the triphasic equilibriun1 is no longer stable. In addition,
`the widening of the biphasic gap occurs at 1nuch higher
`values of X1i as compared to rods \vithout side chains (cf.
`Figure 2a; note the different scales of the ordinate in Figure
`2a-d). Hence, the wide biphasic regime is expected to
`appear at a much lower temperature. Another feature of
`interest is the narrowing of the two-phase region with
`increase of zm at constant x. For a thermal conditions this
`is shown in more detail in Figure 3a.
`Here the volume fractions up and up' in the respective
`phases are plotted against the volume fraction vs' of the
`side chains in the anisotropic phase. It is evident that
`appending of even short chains results in a strong nar·
`rowing of the biphasic gap. This effect is most pronounced
`at higher axial ratios x where the two-phase region becomes
`negligible if v/ > u/. Figure 3b shows the respective
`volume fractions uz and u/ of the rigid core of the polymer.
`As is obvious from this plot, the presence of flexible side
`chains changes the width of the biphasic gap but leaves
`the 1nagnitude of Vz nearly unaffected. Clearly, as in Figure
`
`Roxane Labs., Inc.
`Exhibit 1022
`Page 005
`
`

`
`Macromolecu(es, Vol. 19, No. 5, 1986
`
`Rodlike Systems with Flexible Side Chains 1371
`
`,,,, 'j' OS
`
`06
`
`x~so
`
`'"'
`Figure 6, i\1aximum volume fraction of the side chains in the
`anisotropic phase at v "' I, i.e., in the neat liquid at athermal
`conditions. 'I'he quanfity x(r\\ denotes the athermal limit being
`the minimum axial ratio to produce a stable nematic phase in
`the absence of nttractive forces (cf, ref 20).
`
`01
`
`OS
`
`03
`
`Oi
`
`Figure 4.. Volume fraction up of the polymer in the is-Otropic phase
`at equilibrium at athcrmal conditions vs. zm/x which gives the
`ratio of the volumes occupied by the side chains and the rigid
`core in the molecule, respectively.
`
`•m '
`
`05
`
`OI
`
`00
`
`:1':
`
`10
`50
`100
`I<'igure 5. Volume fraction Up of the polymer in the isotropic phase
`at equilibrium under athertnal conditions vs. x, the axial ratio
`of the rigid core.
`
`l(
`
`311, the difference between vx and v/ is vanishing when vP
`= Vx + v, is going to unity, i.e., when the 'vhole system is
`filled with polymer. Since at athermal conditions v.x is
`nearly independent of zm, the total volume fraction of
`polymer vE. in the isotropic phase should increase linearly
`with zm. Figure 4 shows the dependence of vP on n(zn1/x)
`to be linear indeed. It has to be noted that VP inust not
`be confused with vP *, the volume fraction for incipience
`of metastable order. 14 Thus, the ordering process among
`the rodlike cores is nearly unaffected by the presence of
`side chains, 'vhich appear t-0 act like a low molecular 'veight
`solvent. This result clearly relates to the fact that one end
`of the flexible chain is fixed at the rigid core, which greatly
`reduces its entropy in solution. At higher polymer con(cid:173)
`centrations the number of configurations available to the
`side chains is becoming smaller. As a consequence of this
`unfavorable entropic effect, the concentration of the
`polymer in the anisotropic phase tends to smaller values
`at increasing values of z1n. Since the isotropic phase is
`located at a tnore dilute regime, the overlap between the
`side chains should be smaller there. Hence, the volume
`fraction v.x of the rigid cores does not change very 1nuch
`upon appending of aide chains. Consequently, the curves
`
`"
`
`s
`
`09
`
`08
`
`x: 25
`
`-----~=100
`
`'
`Figure 7. Order parameters (er. eq 29) at different axial ratios
`x of the rigid core vs. zm/x denoting the ratio of the volume
`occupied by the flexible side chains and t.he rigid core in the
`molecule, respectively.
`
`J ..!!!!.
`
`describing the dependence of u\' on x ere running nearly
`parallel to each other, the deviations at lower values of tho
`axial ratio x notwithstanding (cf. Figure 5.) The situation 1
`of course, is changed drastically if the side chains are al·
`lowed to n1ove freely. As has been demonstrated by Flo(cid:173)
`ry,25 this leads to a marked segregation of the rods and the
`random coils between the ordered and the isotropic phase.
`As already discussed above, vP reaches unity at a certain
`volume fraction of side chains (v,)". This quantity, which
`may be taken as a 1neasure of the maximum volutne
`fraction to be tolerated by a nematic phase, is plotted
`against x in Figure 6. In the lin1it of high axial ratios x,
`(01)"' tends to become unity. At low x (v8)"" is zero at Xciit
`= 6.417, which is the minimum axial ratio necessary for
`the occurrence of a nematic phase in the neat liquid. w In
`the vicinity of Xcr1t the dependence of (u1)'" on x is rather
`steep, indicating that an ordered phase composed of small
`rods can only tolerate a small volume fraction or side
`chains.
`The foregoing considerations only hold true for athennal
`conditions. If Xi increases, the biphasic gap is widened,
`as is obvious from Figure 2b-d. Especially if zm is of the
`order of x, vP becomes very small (cf. Figure 2d). However,
`the dense nen1atic phase at high values of Xi is less con(cid:173)
`centrated in the presence of side chains than compared to
`perfectly rigid rods. This is due to the entropic repulsion
`
`Roxane Labs., Inc.
`Exhibit 1022
`Page 006
`
`

`
`1372 Rallauff
`
`" p
`Po
`
`--~~?; l'm
`
`-30
`-10~
`
`05
`
`OI
`
`-------.-------r--r-~-~
`
`0 2
`
`04
`
`06
`
`Figure 8. Reduced vapor preseure p/p0 vs, volume fraction of
`solv~nt for an axial ratio of the rigid core x =< 100, The quantity
`zm 1s the product of z, the number of side chains per rigid core,
`and m, the number of segments per side chains,
`
`of the flexible side chains discussed above,
`All results n1entioned above suggest that the side chains
`may be compared to an ordinary solvent to a certain ex(cid:173)
`tent. However, as is de1nonstrated in Figure 7, the order
`parameUlr in the nematic phase (cf. eq 29) is lowered \Vith
`increasing zm. Hence, as opposed to an ordinary solvent
`consisting of one segment, the flexible side chains interfere
`with the ordering process of the rodlike core.
`Finally, we discuss the alterations effected by the
`presence of side chains at very high polymer concentra·
`tions. For this purpose the reduced vapor pressure pf Po
`of rods of axial ratio 100 with side chains characterized by
`zn1 is plotted vs. the volume fraction of solvent in Figure
`8 (athermal conditio11s). The concentratio11 range is re·
`stricted to those \Vithin in which the system is homoge·
`neous, the single phase being anisotropic. Even at small
`ratios zm/x the resulting Pf Po deviates strongly from its
`respective value at zm = 0. \Vith zm of the order of x the
`reduced vapor pressure is fully governed by the side chains.
`If zm is becoming larger and larger (zm > x), the results
`of the present treatment coincide with those obtained by
`Flory and Leonard within the frame of their side·chain·
`mixing niodeJ. 10
`Having explored the various consequences of side chains
`on the ordering transition of rigid rods, we now turn to a
`comparison with experimental results. As pointed out in
`the beginning, stiff polyesters with appended flexible alkyl
`chains6•7 seem to be the most promising systems for a
`quantitative test of the model developed herein. Unfor·
`tunately, the few data available on these polyesters only
`refer to the thennotropic behavior. At present, certain
`polypeptides exhibiting a stiff helical core in suitable
`solvents appear to be the best candidates for a test of
`theory. The cholesteric structure of these solutions does
`not hamper a comparison \vith theory since the pitch is
`much larger than typical molecular dimensions. To the
`author's knowledge Flory and Leonard10 were the first to
`point out that the solution behavior of a number of helical
`polypeptides like poly(')-.benzyl glutamate) stems from the
`presence of flexible side chains. 1-'he thermodynamics of
`these polymers has bee11 the subject of numerous studies
`
`J.facromolecules, Vol. 19, 1''o. 5, 1986
`
`since, especially by \V. G. Miller a11d co-\vorkers (ref 11,
`12, 26, 27, and further literature cited therein). Presently,
`a number of excellent reviews on the subject are availa·
`ble. n.~3-3t Thus it suffices to focus only on the features
`relevant for the present purpose. For a comparison of
`theory \vi th experimental data one has