1366
`
`Macromolecules 1986, 19, 1366-1374
`
`by the temperature change of a few degrees near Tc of the
`liquid crystal. The viscosity of the nematic phase of EBBA
`is in the same range as that ofliquid 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 thermal motion 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 multi(cid:173)
`component bilayer film. Discontinuous jump of P at Tc
`is ascribable to greater gas mobility in the liquid-crystalline
`phase of the hydrocarbon component, and the jump be(cid:173)
`comes larger as the hydrocarbon domain is enlarged.
`Concluding Remarks
`Multicomponent bilayer membranes can be immobilized
`in the form of PV A composite films. The DSC and XPS
`data indicate that the hydrocarbon and fluorocarbon bi(cid:173)
`layer components are phase-separated and that the fluo(cid:173)
`rocarbon component is concentrated near the film surface.
`These component distributions produce favorable effects
`on permselectivity of 0 2 gas. The selectivity (P0 ,/ PN,) is
`apparently determined by the surface monolayer (or layers
`close to the surface) of the fluorocarbon component, and
`the permeability is promoted by the presence of large
`domains of the fluid (in the liquid-crystalline state) hy(cid:173)
`drocarbon bilayer.
`
`Acknowledgment. We extend our appreciation to the
`Asahi Glass Foundation for Industrial Technology for fi(cid:173)
`nancial support.
`Registry No. 1, 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 2, 7727-37-9.
`
`References and Notes
`(1) Nakashima, N.; Ando, R.; Kunitake, T. Chem. Lett. 1983, 1577.
`(2) Shimomura, M.; 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) Kunitake, T.; Higashi, N.; Kajiyama, T. Chem. Lett. 1984, 717.
`(6) Takahara, A.; Kajiyama, T., personal communication.
`(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. Bunsen-Ges. Phys.
`Chem. 1981, 85, 789.
`(ll) Barrer, R. M.; Skirrow, G. J. Polym. Sci. 1948, 3, 549.
`(12) Hoffman, S. "Depth Profiling" in Practical Surface Analysis;
`Briggs, D.; Seak, M.P., Eds.; Wiley: New York, 1983.
`(13) Brundle, C. R. J. Vac. Sci. Technol. 1974, 11, 212.
`(14) Kajiyama, T.; Nagata, Y.; Washizu, S.; Takayanagi, M. J.
`Membr. Sci. 1982, 11, 39.
`(15) Kajiyama, T.; Washizu, S.; Takayanagi, M. J. Appl. Polym.
`Sci. 1984, 29, 3955.
`(16) Washizu, S.; Terada, 1.; Kajiyama, T.; Takayanagi, M. Polym.
`J. (Tokyo) 1984, 16, 307.
`(17) Kajiyama, T.; Washizu, S.; Ohmori, Y. J. Membr. Sci. 1985, 24,
`73.
`
`Phase Equilibria in Rodlike Systems with Flexible Side Chainst
`
`M. Ballauff
`Max-Planck-Institut fur Polymerforschung, 65 Mainz, FRG. Received 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 z, the number of side chains per
`rod, and m, the number of segments per side chain. Soft intermolecular 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 athermal 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 phases
`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 become less concentrated with increase of the product of z and m. The deductions of the treatment presented
`herein compare favorably with results obtained on lyotropic solutions of helical polypeptides like poly("(-benzyl
`glutamate).
`
`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
`melting point of typical aromatic polyesters exhibiting a
`mesophase often exceeds 500 °C, 2 which makes the pro(cid:173)
`cessing of these materials by conventional methods very
`difficult. In order to reduce the melting temperature of
`thermotropic polymers, 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. 4 Alternatively,
`sufficient solubility and lower melting point may be
`
`t Dedicated 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(cid:173)
`tions have been made 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 deriva(cid:173)
`tives exhibit thermotropic as well as lyotropic behavior. 8•9
`The solution properties of poly(-y-benzyl glutamates)
`(PBLG) are directly influenced by the presence of flexible
`side chains, as has been pointed out by Flory and Leo(cid:173)
`nard.10 The thermodynamics of PBLG in various solvents
`has been investigated theoretically as well as experimen(cid:173)
`tally by Miller and co-workers,11•12 leading to the conclusion
`
`0024-9297/86/2219-1366$01.50/0
`
`© 1986 American Chemical Society
`
`

`
`Macromolecules, Vol. 19, No.5, 1986
`
`Rodlike Systems 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
`fj. If there is no
`follows as the number of vacancies n0 -
`correlation between the adjacent rows parallel to the axis
`of the domain, the expectation of a vacancy required for
`the Yi+ 1 - 1 first segments of the remaining submolecules
`fj) I n0• If a
`is the volume fraction of the vacancies (n0 -
`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 submolecule 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
`fj) I (n0 -
`fj + :~:::/Yi + zmj). Inas(cid:173)
`of the side chains (n0 -
`much as we assume that there is no correlation of the
`conformation of the side chains with the orientational
`order (see above), the expectancy of a vacancy for the zm
`segments of the side chains is given by the volume fraction
`fj)ln 0• It follows that
`of empty sites (n 0 -
`vi+1 = (no - fj) X
`
`.no~fi
`(no~firj+e1(
`.)X-Yj+l x
`no - !J + LYi + zmJ
`( no~fi rm (1)
`
`or
`
`Vj+1 = (no- fj)f(no- xj + L;yyjwxn01-Yjwzm
`With negligible error we may write
`
`j
`
`(2)
`
`j+1
`(no- x(j + 1) + Ly)!
`fj)!
`(n 0 -
`Vj+1 = no - f(j + 1)! -------,j---no1-Yj+!-zm (3)
`(n0 - xj + L;y)!
`The results deduced by the lattice treatment are ex(cid:173)
`pected to become less secure at greater disorientations, i.e.,
`at larger values of y I x. For the ordered phase under
`consideration here, this problem should be of minor im(cid:173)
`portance, however. As usual,20 the configurational partition
`function ZM may be represented as the product of the
`combinatorial part Zcomb• the orientational part Zorient• and
`the intramolecular or conformational part Zconf:
`
`(4)
`Since the configuration of the side chains is assumed to
`be independent of the order of the respective phase (see
`above), Zconf may be dismissed in the subsequent treat(cid:173)
`ment. The combinatorial part is related to vi by
`
`1 )n,
`
`Zcomb =
`
`(
`
`- 1
`n •.
`
`ITvj
`
`(5)
`
`(6)
`
`and thus follows as
`(n0 - n.x + yn.)!
`)
`( _
`z =
`n n, 1-y-zm
`n 'n'
`0
`where n1 = n0 - fnx denotes the number of solvent particles
`and nx the number of polymer molecules. As usual, y is
`the average value of the disorder index y defined by
`
`comb
`
`X' 1•
`
`with nxy being the number of molecules whose rigid core
`is characterized with regard to orientation by y. Following
`
`(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 model. In this work we present a comprehensive
`treatment of the thermodynamics of stiff rods with ap(cid:173)
`pended flexible side chains. As in the theory of Wee and
`Miller,12 it is based on the lattice model originated by
`Flory,13·14 which previously has been shown to account for
`all experimental facts obtained from suitable model sys(cid:173)
`tems.15-19 Special attention is paid to the form of the
`orientational distribution function, as expressed in terms
`of the Flory-Ronca theory20.21 or in the 1956 approximative
`treatment.13 In order to treat lyotropic systems soft in(cid:173)
`termolecular 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 treatment
`of membranes22 and semicrystalline polymers,23 are not
`taken into account in the present treatment. It is assumed
`that the distances between 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 molecules
`with flexible side chains appended regularly (cf. Figure 1).
`As is customary, we subdivide the lattice into cells of
`linear dimensions equal to the diameter of the rodlike part
`of the particle. Thus the number of segments comprising
`this part is identical with its axial ratio, henceforth denoted
`by x. For simplicity, 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 X 80Jvent = 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 xly segments (cf.
`Figure 1). The relation of y to the angle 1/; of inclination
`to the axis of the domain will be discussed later in this
`section.
`The calculation of the configurational partition function
`rests on the evaluation of the number Vj+ 1 of situations
`available to the j + 1 molecule after j particles are intro(cid:173)
`duced. Let n0 denote the total number of lattice sites and
`f = x + mz the overall number of segments per molecule.
`Pursuant to the calculation of Vj+1, it is expedient to in(cid:173)
`troduce the rigid part first. Doing this we proceed along
`
`

`
`1368 Ballauff
`
`Flory and Ronca,20 the orientational part of Zorient becomes
`uw n )n'Y
`Zorient =II _nY x
`(
`xy
`y
`
`(8)
`
`As in ref 20, wy denotes the a priori probability for the
`interval of orientation related toy, and u is a constant.
`Introducing Stirling's approximation for the factorials, eq
`4 together with eq 5 and 6 gives
`
`Macromolecules, Vol. 19, No. 5, 1986
`
`of the solute particle, and 3 to the flexible side chains. The
`quantities Xij are given as usual 24 by
`Xij = ZcD.wijxjjRT
`
`(17)
`
`where x j represents the number of segments per interacting
`species, Zc is the coordination number, and D.wij (=wij-
`0.5(wii + Wjj)) denotes the change in energy for the for(cid:173)
`mation of an unlike contact. Since the volume fraction v.
`of the side chains depends on Vx through v. = uxzmjx, we
`have
`
`where the quantities v1 = ntfn0 and Vx = nxxfn0 denote
`the volume fractions of the solvent and of the rigid core
`of the polymer, respectively.
`For evaluating the orientational distribution we take the
`last molecule inserted into the lattice as a probe for the
`orientational order at equilibrium (see ref 20). Hence,
`equating j + 1 in eq 2 to n., we obtain
`
`( 1 - Vx( 1 - ~) r (10)
`
`Vy -
`
`which immediately leads to
`nxy/nx = {1-1wy exp(-ay)
`with the quantity a defined by
`
`a = -ln [ 1 - vx( 1 - ~) ]
`
`and
`
`{ 1 = Lwy exp(-ay)
`y
`
`(11)
`
`(12)
`
`(13)
`
`Alternatively, eq 11 may be derived by variational
`methods. 20 Note that the orientational distribution
`function nxy/ nx 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, Zorient follows as
`Zorient = nx[ln {f1 u) + ay]
`and, upon insertion into (9)
`
`(14)
`
`-ln Z = n1 ln v1 + nx ln ~ - (n0 - nxx) ln [ 1 -
`vx( 1 - ~)] - nx(l - y- zm) - nx In (flu) (15)
`
`For examining the effect of small interactions between the
`solute particles, the heat of mixing tJfM is incorporated
`into the expression for the free energy. If it is assumed
`that randomness of mixing is preserved at all degrees of
`orientation, tJfM 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 familiar in(cid:173)
`teraction parameters x12, x13, and x23, tJfM becomes
`tJIM = X12n1v2 + X13n1vs + X2sn2v3
`(16)
`where the index 1 refers to the solvent, 2 to the rigid part
`
`with
`
`and
`
`zm
`X1 = X12 + ~X13
`
`zm
`X2 = ~X23
`
`(18)
`
`(19)
`
`(20)
`
`The chemical potentials are derived from eq 7 and 18 with
`the stipulation that at orientational equilibrium
`
`ln ZM )
`(nxy/nx)
`
`(
`
`n~on, = O
`
`Hence
`
`11J.L1' jRT = ln v1 + v.' + _:_(y- 1) +a+
`
`VI
`
`X
`
`]
`vx' zm
`(vx' + vs')x1 - -;- ~X2
`
`(21)
`
`vx'
`[
`
`and
`
`11J.Lx' jRT = ln v;'- { v1' + v;') + f- vx'{ 1- ~) +
`zma- ln ({1u) + ( 1- vx'~ )( v1'xx1 + v/: 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
`y to x, are
`11J.Lt/RT =
`
`ln V1 + V8 + ux( 1- ~) + vx[ (vx + v.)xl- ~ z: X2]
`11J.Lx/RT = ln ~ + f + t( V1 + ~)- ln u +
`( 1 - Vx~ )( V1XX1 + Vx z: X2) (24)
`
`(23)
`
`For evaluating the composition of the conjugated phases
`by means of the equilibrium conditions
`11J.Ll = 11J.Ll'
`11J.L2 = 1111-2'
`the relation between y and the angle of inclination has to
`be specified. Since the orientational distribution as rep(cid:173)
`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 treatment of the
`lattice model. 20 Thus the disorder index y is represented
`by
`
`(25)
`
`(26)
`
`

`
`Macromolecules, Vol. 19, No.5, 1986
`
`Rodlike Systems with Flexible Side Chains 1369
`
`(27)
`
`(28)
`
`(29)
`
`4
`.
`,/,
`y = -x sm 't'
`11'
`In orientational equilibrium its average value follows as
`4
`Y = -xfdfl
`11'
`and the order parameter s becomes
`s = 1 - o/Ja/!1
`with the integrals {p being given by
`f'lf/2
`{p = Jo
`
`sinP '/; exp(-a sin'/;) M
`
`(30)
`
`The factor a relates to a (cf. eq 12) by
`4
`a= -ax
`11'
`For values of a» 11' /2, the integrals {p may be developed
`in series, yielding
`
`(31)
`
`a4
`
`fl = _!_ + (!)( _!_)
`a2
`{2 = ~ + (!)( _!_)
`a3
`as
`fa = ~ + 0( _!_)
`
`a4
`
`a6
`
`(32)
`
`(33)
`
`(34)
`
`In this asymptotic approximation20 the average disorder
`parameter y directly follows from
`2
`2
`y =;; = -ln [1- ux(l- yjx)]
`Thus, we obtain for the chemical potentials eq 23 and 24
`the expressions
`ux'
`2
`6.f..L1' jRT = ln u1' + u.' + -(y- 1) +-= +
`y
`X
`
`(35)
`
`ux' [ (ux' + us')x1 - u;' z: X2 ]
`
`(36)
`
`where C = 2 ln (11'e/8). Further simplification can be
`achieved by omitting the small constant C and equating
`a to x (1956 approximation; cf. ref 11). In order to obtain
`the volume fraction of the polymer in the coexisting phases,
`trial values of ux' and a (cf. eq 12) are chosen which serve
`for the numerical evaluation of the integrals {p 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 terms. With the {p being known, y follows from (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 ux, from
`which an improved value of ux' is accessible via eq 25. The
`entire calculation is repeated with a new value of ux' until
`compliance with all conditions for equilibrium is reached.
`When the 1956 approximation is used, the same scheme
`
`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 ux' and ux'' of two an(cid:173)
`isotropic phases in equilibrium can be evaluated by a
`similar numerical solution of (25) and (26) after insertion
`of (21) and (22) together with (28). Since this calculation
`is more sensitive toward the orientational distribution
`function, the exact treatment of Zorient has to be used.
`Results and Discussion
`Parts a-d of Figure 2 show the phase diagrams resulting
`for rigid rods of axial ratio x = 100 with flexible side chains
`characterized by the number of side chains z per rod and
`the number of segments m per side chain.
`From the theoretical treatment it is evident that only
`the product zm matters. Thus the polymer molecules
`under consideration here are fully determined by x and
`zm. Since the segments of the side chains are assumed to
`be isodiametric with the segments of the rigid core, the
`quantity zm / 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 x1 (see eq 19) measuring
`the interaction of the solvent and the rigid core will suffice
`to take into account intermolecular forces. The second
`parameter 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 form of the orientational distribution (see eq
`36-38). The result given herein was obtained by resort to
`the treatment of the lattice model originated by Flory and
`Ronca. 20 A comparison of both methods shows that the
`main features of the diagram are rather insensitive toward
`the respective form of the orientational distribution
`function nx)nx. However, the stability of the equilibria
`between two nematic phases is profoundly influenced by
`nxy/nx. This fact becomes even more important 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
`nematic-nematic equilibrium in the frame of the 1956
`approximation. From these results it is obvious that the
`exact treatment of nx /nx 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 from a comparison of Figure 2, part a,
`with parts b-d, there is a marked influence of the side
`chains on the phase behavior of rigid rods. We first discuss
`Figure 2, part a, 2a together with part b referring to zm
`= 0 and zm = 2, respectively. Here uP 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 x1 = 0 the soft in(cid:173)
`teractions between the rods and the solvent are null,
`corresponding to athermal conditions. The narrow hi(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 n,n may occur. Further in(cid:173)
`crease of Xt> equivalent to a lowering of temperature, leads
`to a triple point where three phases are in equilibrium.
`Raising x1 to still higher values is followed by a dilute
`isotropic phase in equilibrium with a dense nematic phase.
`
`

`
`1370 Ballauff
`
`Macromolecules, Vol. 19, No.5, 1986
`
`012
`x,
`0 I
`
`009
`
`007
`
`005
`
`003
`
`001
`
`I
`I
`
`'
`I
`I
`I
`
`\I
`
`I
`
`• N
`
`X
`
`:100
`zm=O
`
`00
`
`02
`
`06
`
`OS
`
`10
`
`10
`
`Vp, Vp'
`
`08
`
`0.6
`
`04
`
`02
`
`b i
`
`00
`00
`
`0.2
`
`I 0
`Yx,V~
`
`08
`
`06
`
`0.4
`
`" "
`0.2
`X =50 ",
`~-----------------x~=-1~00~--'~
`~-.-~.--.-,--,---,--,-~
`02
`04
`0.6
`08
`1.0
`00
`
`"
`
`I
`
`i
`I
`
`Figure 3. (a) Volume fraction vP and vp' of the polymer in the
`coexisting isotropic and anisotropic phase, respectively, at athermal
`conditions vs. the volume fraction v,' of side chains in the an(cid:173)
`isotropic phase. (b) Volume fraction Vx and vx' of the polymer
`in the coexisting isotropic and anisotropic phase, respectively,
`at a thermal 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 x1 and its 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 zm
`= 0. Thus, 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 more obvious
`when higher values of zm (Figure 2c,d) are used. Here,
`the triphasic equilibrium is no longer stable. In addition,
`the widening of the biphasic gap occurs at much higher
`values of x11 as compared to rods without 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 athermal 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 us' of the
`side chains in the anisotropic phase. It is evident that
`appending of even short chains results in a strong nar(cid:173)
`rowing of the biphasic gap. This effect is most pronounced
`at higher axial ratios x where the two-phase region becomes
`negligible if u.' > u,'. Figure 3b shows the respective
`volume fractions ux and ux' 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 magnitude of ux nearly unaffected. Clearly, as in Figure
`
`I
`
`• N
`
`"100
`X
`zm =2
`
`01
`
`OJ
`
`05
`
`07
`
`---~
`
`X
`
`:100
`ZIT':30
`
`x,
`0 l5j
`
`0 l(i
`
`I i
`
`0121
`
`0 I
`
`,
`1
`009: I
`
`007,
`
`005
`
`x,
`
`06
`
`as
`
`04
`
`03
`
`02
`
`01-
`
`00
`
`02
`
`04
`
`~--~·---------,
`
`d I
`
`X,
`
`12
`
`:1QQ
`X
`zm= 100
`
`02
`
`06
`
`OS
`
`10
`Yp.YP
`
`Figure 2. Phase diagrams of rigid rods immersed in a solvent;
`x is 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 x1•
`The quantities vP and Vx denote the volume fraction of the total
`polymer and its rigid core, respectively.
`
`Several features command attention when Figure 2, parts
`a and b, are compared. Going from zm = 0 (perfectly rigid
`rods) to zm = 2 narrows considerably the region where to
`nematic phases are coexisting. If zm > 4, this region be-
`
`

`
`Macromolecules, Vol. 19, No.5, 1986
`
`Rodlike Systems with Flexible Side Chains 1371
`
`lv5 I
`
`08
`
`06
`
`04
`
`0. 2
`
`00
`
`10
`50
`30
`100
`Figure 6. Maximum volume fraction of the side chains in the
`anisotropic phase at uP = 1, i.e., in the neat liquid at athermal
`conditions. The quantity Xcnt denotes the athermallimit being
`the minimum axial ratio to produce a stable nematic phase in
`the absence of attractive forces (cf. ref 20).
`
`lll.
`X
`Figure 4. Volume fraction Up of the polymer in the isotropic phase
`at equilibrium at athermal conditions vs. zmjx which gives the
`ratio of the volumes occupied by the side chains and the rigid
`core in the molecule, respectively.
`
`Vp
`
`as
`
`0.1
`
`}
`
`00
`
`10
`50
`100 X
`Figure 5. Volume fraction uP of the polymer in the isotropic phase
`at equilibrium under athermal conditions vs. x, the axial ratio
`of the rigid core.
`
`3a, the difference between Vx and v,' is vanishing when vP
`= vx + u. is going to unity, i.e., when the whole system is
`filled with polymer. Since at athermal conditions Vx is
`nearly independent of zm, the total volume fraction of
`polymer v.£ in the isotropic phase should increase linearly
`with zm. l<'igure 4 shows the dependence of vP on n(zmjx)
`to be linear indeed. It has to be noted that Vp must 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, which appear to act like a low molecular weight
`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 zm. Since the isotropic phase is
`located at a more dilute regime, the overlap between the
`side chains should be smaller there. Hence, the volume
`fraction Vx of the rigid cores does not change very much
`upon appending of side chains. Consequently, the curves
`
`1.0
`
`s
`
`0.9
`
`08
`
`3 .l!!l
`X
`Figure 7. Order parameters (cf. eq 29) at different axial ratios
`x of the rigid core vs. zmjx denoting the ratio of the volume
`occupied by the flexible side chains and the rigid core in the
`molecule, respectively.
`
`describing the dependence of vP on x are running nearly
`parallel to each other, the deviations at lower values of the
`axial ratio x notwithstanding (cf. Figure 5.) The situation,
`of course, is changed drastically if the side chains are al(cid:173)
`lowed to move 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 measure of the maximum volume
`fraction to be tolerated by a nematic phase, is plotted
`against x in Figure 6.
`In the limit of high axial ratios x,
`(v.)"' tends to become unity. At low x (v.)"' is zero at Xcrit
`= 6.417, which is the minimum axial ratio necessary for
`the occurrence of a nematic phase in the neat liquid.20 In
`the vicinity of Xcrit the dependence of (v.)"' on x is rather
`steep, indicating that an ordered phase composed of small
`rods can only tolerate a small volume fraction of side
`chains.
`The foregoing considerations only hold true for athermal
`conditions. If x1 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 nematic phase at high values of x1 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
`
`

`
`--::::::::::;::::;;;;.;;;?~ j' zm
`
`-30
`-100
`
`X : 1QQ
`
`1372 Ballauff
`
`10
`£....
`Po
`
`0.5
`
`01
`
`00
`
`0.2
`
`04
`
`0.6
`
`Figure 8. Reduced vapor pressure p /Po vs. volume fraction of
`solv.ent for an axial ratio of the rigid core x = 100. The quantity
`zm IS 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 mentioned above suggest that the side chains
`may be compared to an ordinary solvent to a certain ex(cid:173)
`tent. However, as is demonstrated in Figure 7, the order
`parameter in the nematic phase (cf. eq 29) is lowered with
`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(cid:173)
`tions. For this purpose the reduced vapor pressure p / p 0
`of rods of axial ratio IOO with side chains characterized by
`zm is plotted vs. the volume fraction of solvent in Figure
`8 (athermal conditions). The concentration range is re(cid:173)
`stricted to those within in which the system is homoge(cid:173)
`neous, the single phase being anisotropic. Even at small
`ratios zmjx the resulting P/Po deviates strongly from its
`respective value at zm = 0. With 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(cid:173)
`mixing model. 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
`chains5•7 seem to be the most promising systems for a
`quantitative test of the model developed herein. Unfor(cid:173)
`tunately, the few data available on these polyesters only
`refer to the thermotropic 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 with 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('y-benzyl glutamate) stems from the
`presence of flexible side chains. The thermodynamics of
`these polymers has been the subject of numerous studies
`
`Macromolecules, Vol. 19, No. 5, 1986
`
`since, especially by W. G. Miller and co-workers (ref 11,
`I2, 26, 27, and further literature cited therein). Presently,
`a number of excellent reviews on the subject are availa(cid:173)
`ble.11·28-31 Thus it suffices to focus only on the features
`relevant for the present purpose. For a comparison of
`theory with experimental data one has to bear in mind that
`the present model only treats molecularly uniform systems.
`However, the deviations due to the finite width of the
`distribution of chain length is expected to be of minor
`importance in the course of a more qualitative discuss