Faraday Symp. Chem. Soc. 1983 18,49-56 Viscoelasticity of Concentrated Solutions of Stiff Polymers BY MASAODOI Department of Physics Faculty of Science Tokyo Metropolitan University Setagaya Tokyo Japan Received 0th August 1903 A solution of rodlike polymers forms a liquid-crystalline phase above a certain concentration. In this paper we derive the constitutive equation for such a system under a weak velocity gradient from the molecular-kinetic theory. The equation agrees with Leslie’s phenomenological equation in the liquid-crystalline phase. The Leslie coefficients are expressed by molecular parameters such as the concentration molecular weight and the order parameter. 1. INTRODUCTION The viscoelastic properties of solutions of rodlike polymers depend strongly on concentration.At very low concentration the solution is nearly a Newtonian fluid with low viscosity. As the concentration increases the viscosity increases sharply and marked viscoelasticity appears. With a further increase in concentration the solution forms a liquid-crystalline phase1 and shows characteristic rheological properties which are quite distinct from those of isotropic solutions. In this paper we discuss how such behaviour may be understood on the molecular level. In particular we discuss the characteristic distinction between the isotropic phase and the liquid-crystalline phase. From the viewpoint of rheology the basic distinction between the isotropic phase and the liquid-crystalline phase is behaviour under low velocity gradients.For isotropic fluids the stress tensor tr for a steady flow is always given by Newton’s law if the velocity-gradient tensor K = (Vu)? is sufficiently small c = ~(K+K+). (1.1) (In this paper the isotropic part of the stress tensor is omitted for simplicity.) Since eqn (1.1) is the first term in a power-series expansion of c with respect to K it is generally correct as far as the stress tensor is an analytic function of K. From eqn (1.1) one can conclude for example that the elongational viscosity is generally three times larger than the shear viscosity at low elongational rate. On the other hand such a relation does not hold in the liquid-crystalline phase. In this case the relation between cr and K becomes singular at K = 0. This is seen in the phenomenological constitutive equation proposed by Leslie2 for the nematic liquid crystals.In nematics the system has uniaxial symmetry around a certain direction specified by a unit vector n called the director. If n is fixed by some external field the relation between cand K is analytic. However if there is no external field the direction of n itself is determined by the velocity gradient tensor K. As a result the relation between e and K becomes non-analytic. The problem discussed here is how such a ‘singular’ constitutive equation is derived from molecular theory. We shall show that Leslie’s phenomenological equation is naturally derived from the molecular kinetic equation and express the phenomeno- logical parameters appearing in the equation by molecular parameters.49 VISCOELASTIC PROPERTIES OF STIFF POLYMERS 2. BASIC EQUATION The polymer we consider here is a rodlike polymer of length L and diameter b(L + b). First we briefly review the static theory for the formation of the liquid- crystalline phase. Let flu) be the orientational distribution function which is the probability that an arbitrarily chosen polymer is in the direction of the unit vector u. According to on sage^,^ the free energy per unit volume of solution is expressed as a function offlu) as d[f]= ckB T(sduflnf+-2‘idu du’p(u u’)flu)flu’)) (2.1) where c is the number of polymers per unit volume k T is Boltzmann’s constant multiplied by the temperature and p(u,u’) is the excluded volume for a pair of polymers in the directions u and u’ D(u u’) is calculated from the interaction energy u(r,u;r’,~’) between the two polymers in the configuration (r,u)and (r’,~’)(r and r’ denoting the positions of the centres of mass) as Jr D(u,u’) = dr{1 -exp [-u(r,u; 0 u’)/kgT]}.For a rigid rod P(u,u’) = 2bL21ux u’(.~ In general B(u,u’) takes minimum when u is parallel (or antiparallel) to u’. The equilibrium distribution function is determined from the condition Sd/Sf = 0 which leads to fO(u) = exp -hCF(U; [f0l)lkB r) (2.3) where C is a constant and I/SCF(u [fJ) = ck T du’p(u,u’)flu’) (2.4) i is termed the mean field potential. Eqn (2.3)is a non-linear integral equation for fo and its solution generally requires the numerical calculation. However if B(u,u’) is approximated as4 cD(u,u’) = c-; U(U*U’)2 (2.5) where U is a parameter proportional to c eqn (2.3)is easily solved as fo(u)= C exp [ -US(~*U)~] (2.4) where n is an arbitrary unit vector and S is the solution of the equation du [(~=n)~-#] exp[-$US(n*u)2] S= (2.7) Cdu exp [-4 US(u n)2] If U is small eqn (2.7)has only one solution S = 0.This corresponds to the fact that the system is isotropic at low concentrations. On the other hand if U is sufficiently large eqn (2.7)has another solution for positive values of S. This solution represents the liquid-crystalline phase in which polymers are oriented towards n. To discuss the viscoelasticity we must know the time-evolution equation for the distribution functionflu t).In dilute solution such an equation is given by Kirkwood et a1.5as (2.8) at M.DO1 51 a where B=UX-(2.9) au is the operator of rotation 5 is the potential of the external field and D,,is the rotational diffusion constant which is related to the rotational friction constant [ro through Einstein’s relation Dro = k T/Cro* (2.10) The friction constant crois calculated by hydrodynamics as5? (2.11) where qSis the viscosity of the solvent.In concentrated solutions eqn (2.8) is modified reflecting various types of interaction between the polymers. Two interactions are particularly important (i) The excluded-volume interaction. As is shown in the static theory the excluded- volume interaction acts by aligning the polymers in the same direction. This effect is taken into account by adding the mean-field potential &cF(u,~u; t)]) to & in eqn (2.8).7-1 O (ii) The entanglement interaction.The constraint that polymers cannot pass through each other becomes a serious restriction for rotational Brownian motion of polymers in a concentrated solution. As a result the rotational motion is significantly slowed down. According to ref. (1 1) and (12) the effective rotational diffusion constant Dr is estimated as Dr = pD,0(cL3)-2 du’JTu; t)fTu’; t)lu x u’I (2.12) where 8’ is a numerical coefficient. Thus the time evolution equation in concentrated solution is obtained as (2.13) Two comments may be added. (a) In eqn (2.13) the hydrodynamic interaction is neglected. For flexible polymers this interaction is known to be important since the hydrodynamic interaction becomes screened as the concentration l4 For rodlike polymers however this effect is not very important because the effect of the hydrodynamic interaction is already weak in dilute solution.In fact the hydrodynamic interaction gives only a small correction to D,,in eqn (2. l6 (b)Although eqn (2.12) predicts correct molecular weight and concentration dependence,I7 its validity has been questioned18-20 since the experimentally obtained value of p is much larger than the naive expectation which is to say that p is of the order of unity.12 A recent careful estimation of j?’was made by Hayakawa,21 who showed that can be of the order of lo3 if the distribution of disengagement times is correctly taken into account.Therefore the discrepancy in the magnitude of Dr is not very large although the agreement between the theory and experiment is not yet complete. If the distribution function is known the stress tensor may be calculated by the procedure discussed in ref. (22). The stress tensor of polymer solutions is generally written as d = aS+aE+aV (2.14) where dS = qs(K+K+) (2.15) VISCOELASTIC PROPERTIES OF STIFF POLYMERS is the stress of the pure solvent and oEand trV represent the contributions of the polymers which are obtained as follows. We consider a hypothetical deformation BE which instantaneously displaces the material point at Y to ~+BE*Y.This deformation changes the free energy from Lcam to LcaLf].The stress oEis related to the change in Lca by this deformation as t7E BE = d[ f '1-dlfl.(2.16) On the other hand oVis given by the hydrodynamic energy dissipation caused by the deformation while the system is deformed. For the rodlike polymers the deformation BEchanges the distributionfas [see eqn (2.1311 f +f' =f-4E-[ux(BE-u)fJ. (2.17) From eqn (2.1) (2.16) a nd (2.17) we get tYE = 3Ck~T(UU-gI) C(U[UX 3(J'&F K)]) (2.18) On the other hand uVand nSare shown to be much smaller than cE.Hence the stress can be evaluated by eqn (2.18). 3. PERTURBATION EXPANSION Eqn (2.13) and (2.18) apply for both the isotropic phase and the liquid-crystalline phase. However a crucial difference appears in the mathematical handling of the equations for the two phases.For simplicity we consider the steady state in a weak velocity gradient. The equation to be solved is &-l+Glfl=O (3.1) (3.2) GW= -9*UX(K'Uf). (3.2) (For simplicity we assume =0.) The operator $is a non-linear operator while G is a linear operator. To solve eqn (3.1) we assume the solution in the following form f =fo+f1+ ... where fois the solution at equilibrium and fl is the first-order perturbation in K. Substituting eqn (3.4) into eqn (3.1) and comparing the first-order term in K we get ~flI+G[fOl= 0 (3.5) where the linear operator H is the first-order perturbation in $ i.e. +f1l-mo mo1 = m11+~(f12)* (3.6) In the isotropic phase fois equal to 1 /4n,and the linear equation (3.5) is easily solved. On the other hand in the liquid-crystalline phase this procedure encounters a difficulty because the equilibrium state foin the liquid-crystalline phase is not unique as the director n is arbitrary.Thus in the perturbation expansion for the liquid-crystalline phase we have to determine both the unperturbed state as well as the perturbed state. This is done as follows Let yAi) and qW be the right-hand-side and the left-hand-side eigenfunctions for the operator H Aw(i) @@i) = -~(z)~(i) = -~(i)@i). (3.7) M.Do1 53 These functions can be assumed to be orthonormal Sij = (@i)l~(j))E du @i)(u)V/~)(U). s We look for the solution of eqn (3.5) in the form fl= C ai yAi). (3.9) i Substituting eqn (3.9) into eqn (3.5) and using eqn (3.8) we get L@)ai= (qVi)lG[fo]). (3.10) This equation has solution if > 0 for all i.However the operator H has a zero eigenvalue. This is a direct consequence of the fact the equilibrium state in the liquid- crystalline phase is continuously degenerate. In fact iffo andfi denote the equilibrium states which have directors nand n’ = n+an,respectively then Sfo =fi -focorresponds to the eigenfunction of zero eigenvalue since fiSfO]= PLfJ-P[fO]= 0. (3.11) Therefore to obtain the steady-state solution we must have (Plmol) = 0 (3.12) for the eigenfunction qVi) corresponding to the zero eigenvalue. This equation determines the unperturbed state fo. 4. THE CONSTITUTIVE EQUATION FOR WEAK VELOCITY GRADIENT We now show that Leslies’s equation is actually derived from the above formulation. Although it is possible to work out eqn (2.13) and (3.12) we take a simplified ap- proachlO here for the purpose of demonstration.Instead of eqn (2.13) we construct an approximate equation for the following quantity Q f(UU-$1) (4.1) which is called the orientational-order parameter tensor. To obtain an equation for Q we multiply both sides of eqn (2.13) by uu-1/3 and integrate over u. Using the Hermitian property of i9 i.e. J du A(u)gB(u) = -J du[9B(u)]A(u) we can write the result as a -Q = -60 Q+ 60 U(Q*(UU) -(UUUU) Q) at By the use of the following decoupling approximation (uuuu) :Q = (uu) (UU) :0 (4.4) (UUUU) K = (UU) (UU) K eqn (4.3) is rewritten aslo a -Q= F+G (4.5) at VISCOELASTIC PROPERTIES OF STIFF POLYMERS where F = -6Dr[(1 -y) Q-U [Q.Q-g(Q Q) I]+ U(Q:Q) Q3 (4.6) G = ;(K + K?) + KOQ + Q xf -#(K :Q) I-2(~:Q) Q. (4.7) If G = 0 the equilibrium solution of eqn (4.5) is written as Q(O) = S(nn-$ I) ('4.8) where S is a solution of the equation -$Us+;Us2 =o1 (4.9) i.e. for U< 3 (4.10) This equation explicitly shows that the liquid-crystalline phase appears for large values of u. To obtain the perturbed solution we write Q = Q(O)+Q(l) and expand the equation F+G = 0 as 0 = F,p(Q) + G,p(Q) = Hap,pv Q$ + G,p(Q(O')+ 8(1c2) (4.1 1) where a,/?(= x,y z)denote the component of the tensors and summation is implied oyer the repeated Greek indices. The matrix Has,pv,which corresponds to the operator H is calculated from eqn (4.6) as Hap,pv= [(1 -y)+$US-$US2 1dapdpv -US(~gpnanV+~,,npnp) -2US2(nang-&p) (npnv-Qdpv).(4.12) The eigenvector @),which corresponds to the eigenfunction qW (u)of zero eigenvalue now satisfies Hpv,ap4g = 0. (4.13) The solution of eqn (4.13) is q5$) = n rnf)+ ngm&Q (4.14) where mci)is an arbitrary vector perpendicular to n.The condition (3.12) is now written as d$ G,g(Q'O') = 0 (4.15) From eqn (4.7) (4.8) (4.14) and (4.19 it follows that [(I -~)~,n~+(1+2S)1~,~ng]rn~) (4.16) = 0. Since rn@) is perpendicular to n this condition is rewritten as [(l-S)~t'n+(1+2S)rc*n]xn=0. (4.17) Eqn (4.17) determines the direction of the alignment in the unperturbed state. This result was already obtained in ref. (10) by a different method. Given the unperturbed state it is easy to calculate the stress tensor.For the potential (2.5) eqn (2.18) is written as t~ = ck T[3Q+(Q*(uu)-(uuuu):Q) U] (4.18) M.DO1 55 By use of the decoupling approximation eqn (4.4) and the steady-state condition F+G = 0 eqn (4.18) is rewritten as So far we assumed that there is no external field. In order to compare the present theory with Leslie's theory we have to introduce an external magnetic field H. The potential & for the magnetic field is given by I/e = -ia,(u*H)2 (4.20) where a = al,-al is the difference in the susceptibility of the polymer for the two cases that u is parallel to H and the case that u is perpendicular to H. Repeating the same calculation we obtain the following equations in place of eqn (4.5) i3 -Q= F+G+M (4.21) at where F and G are given by eqn (4.6) and (4.7)' and M is defined by M=-Da ['(HH-:PI) -$H2Q+HH*Q+Q-HH-'(HH Q)l -2(HH Q)Q].kBT (4.22) The stress tensor is calculated from eqn (2.18) and (4.20) as ck T[F(Q)+M(Q)] +;Zcaa(HH* g = -~ Q-Q.HH) 2Dr CkBT --G(Q) +ica,(HH*Q-Q*HH) 2Dr c =-ck T &(K+K~) +S(~mnn+nn.~t)-22S2(~:nn)nn] +-a S(H*n)(Hn-nH). 2Dr 2 (4.23) Eqn (4.15) is now replaced by d$[ Gap(0"))+Map(Q(O))] = 0 (4.24) which leads to (4.25) Introducing the symmetric and antisymmetric tensors A = ~(K+KT); Q = ~(Ic-K~) (4.26) and the 'molecular field' h = ca S(H*n)H (4.27) we can rewrite eqn (4.23) and (4.25) in the form of the Leslie equation o = a,(nn:A)nn+a,nN+a,Nn+a,A+a,nn.A+a,A*nn (4.28) and nx(h-y,N-y2A*n)=0 (4.29) dn where N = -fl*n.(4.30) dt VISCOELASTIC PROPERTIES OF STIFF POLYMERS The coefficients are obtained as a = -2S2q:a2= -S (I+-;fs)if:a3=-s (I-2:Sb a = f(l -S)q as = 2Sq a = 0 6s2 -q:y2 = -2sq (4.31) Y’-2+s where q = ck T/2& Leslie’s relation y = a3 -a2 y2 = a -a and Parodi’s relation22y2 = a3+a2are satisfied. Eqn (4.31) coincides with the result already obtained by Marrucci21 based on eqn (4.21). However the reason for this coincidence is not clear since Marrucci’s calculation is for the special case that the director is determined by the magnetic field. Note that the constitutive equations (4.28) and (4.29) also hold for the isotropic solution if S = 0 eqn (4.29)is automatically satisfied and eqn (4.28)reduces to b=-2ck T (K +K+).(4.32) 3Dr Eqn (4.9) and (4.31)predict the chracteristic behaviour of rodlike polymers i.e. the concentration dependence of the shear viscosity and the relative magnitude of the Leslie coefficients.lO* However due to mathematical approximations such as eqn 24 (2.5)and (4.4) the result it perhaps not very accurate. More accurate calculations are now being and will be published soon. This work was supported by a Grant-in-Aid for Scientific Research by the Japanese Government. P. G. de Gennes The Physics of Liquid Crystals (Clarendon Press Oxford 1975). * F. M. 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