Colligative Properties of Helical Polyelectrolytes

Transcription

1 Vol. 12, No. 3, May-June 1979 Colligative Properties of Helical Polyelectrolytes 515 Colligative Properties of Helical Polyelectrolytes Jeffrey Skolnick* Departent of Cheistry, Yale University, New Haven, Connecticut Received May 26, 1978 ABSTRACT: In the context of a continuu odel, the excess electrostatic free energy, F, of helical charge distributions has been obtained..more specifically, we calculate F,,, of cy, DNA-like, and infinite pitch helical lines of charge ebedded on the surface of a low dielectric cylinder. The deviation of F,,, fro the uniforly charged cylinder value is evaluated as a function of screening length. The deviation is sall for DNA, saller for the a helix, and significant for the helix of infinite pitch. Moreover, eploying the ideas of Manning, the colligative properties of such polyelectrolyte solutions have been obtained. The deviation fro a line of charge in bulk solvent odel is sall at low salt concentration but becoes appreciable at higher ionic strengths. In a recent paper, hereafter designated paper 1, the Debye-Huckel equation was solved for the electrostatic interaction energy of point charges on the surface of a dielectric cylinder iersed in salt water; ajor deviations fro the interaction in bulk solvent were found. We then applied the analytical results to calculate that portion of the potential, As, of helical charge distributions which depends on salt concentration. For a and DNA-like helical charge distributions, the doinant contribution to A s is the potential characteristic of a uniforly charged cylinder. In toto, the treatent of paper 1 indicated the large and at ties surprising effect a low dielectric, salt-excluding cylinder can have on the electrostatic potential. The present work deals with the calculation of A@ and various colligative properties of a, DNA-like, and infinite pitch helical charge distributions. Our approach is in the spirit of Manning,2-4 and by analogy to siple electrolyte theory, the derived quantities have been designated extending Manning colligative properties. The reader is referred to paper 1 for a brief review of the 1iterature.l As a first approxiation, it sees reasonable to view the helical polyion as a thin helical line of charge ebedded on the surface of a low-dielectric, obile, ion-free cylinder. a helices are represented by a single helical line of charge; DNA-type double helices are odeled by two helical lines of charge 180 out of phase with each other. When the pitch of the helix is infinite, the helix becoes a line of charge ebedded on the surface of a low dielectric cylinder. Such a odel is perhaps appropriate for linear, nonhelical polyelectrolytes3 and is therefore exained. A quantity of priary interest is the difference in reversible work required to charge up a given charge distribution in the presence and absence of salt, F,,,,,,. The difference between the electrostatic free energy of a unifor charge distribution and the helical line is found to be rather sall for either the a helix or for DNA but increases with increasing pitch and increasing salt concentration. Possibly, the difference ight be significant for DNA at high salt concentrations. Although the dielectric cylinder exerts a drastic influence on the interaction between point charges, the effect has a different sign depending on whether the charges reside on the sae or opposite sides of the cylinder, and there appears to be a significant cancellation for a helical charge distribution. However, over a considerable range of ionic strengths, ajor deviations in FeXCeSS of the infinite pitch helix fro both a line and unifor charge odel are observed. As one would expect, the deviations becoe greater with increasing salt concentration. * Address correspondence to this author at Bell Laboratories, Murray Hill, New Jersey Since the deviations fro ideality of polyelectrolyte colligative properties can be related to (afexcess/ak)t,v, we deterine this quantity for the a helix, line of charge, and DNA double helix. In each case, we delineate the conditions under which the line of charge in bulk solvent is the appropriate liiting case and when corrections becoe necessary. Finally, we forulate extended Manning colligative properties for the osotic and activity coefficients of a helical, DNA-like helical, and infinite pitch helical polyelectrolytes. The Excess Electrostatic Free Energy and Colligative Properties of Helical Charge Distributions A. Description of the Model. In view of the quite large effect of a cylinder on the interaction between point charges on its surface,l especially at high salt concentration, we have exained the self-energy of helical distributions of charges. Specifically, one of the questions put in this section is whether the effect of varying salt concentration on the self energy differs between the helical array and a unifor charge distribution. The self energy is independent of the internal dielectric constant for the unifor distribution. Having isolated that portion of the self energy which depends on salt concentration, Fexcess, we then consider the influence of the helical charge distribution on the colligative properties of the polyelectrolyte solution. For a and DNA-like helical lines, our nuerical work indicates that the difference in electrostatic free energy between the helical and unifor distributions is rather sall. As Manning s liiting laws are easy to apply and represent a well-defined reference ~ tate,~-~ it sees worthwhile to exaine the predictions of Manning theory at finite salt concentration. Otherwise stated, we postulate the following at finite salt concentration: (I) If.$ > 1 counterions condense on the polyion to reduce E to one. Here, E = q2/d2kbtb, where q is the protonic charge, kb is Boltzann s constant, Tis the absolute teperature, b is the linear distance between charges in the configuration of axiu extension, and D2 is the bulk solution dielectric constant. A 1:l supporting electrolyte is assued to be present. (This is Manning s assuption D.)2 (11) We ay treat the uncondensed ions in the Debye-Huckel approxiation. (This is Manning s assuption E.)2 (111) We neglect interactions between different polyions in the calculation of colligative properties. (IV) The helical polyelectrolyte is assued to be infinitely long. (V) The helical polyelectrolyte is a salt-excluding, low-dielectric cylinder iersed in bulk solvent; the helical lines of charge reside on the surface. Plausible rationalizations for assuptions I and I1 are as follows: First and foreost, we wish to define a siple /79/ $01.00/ Aerican Cheical Society

2 516 Skolnick Macroolecules reference syste valid at finite salt concentration. Furtherore, in soe recent work, Manning4 quotes a wealth of experiental data that supports the plausibility of assuption (I). For exaple, H. Skielan, J. M. A. M. van der Hoeven, and J. C. Leyte5 perfored soe NMR studies on the sodiu salts of polyphosphate, polyacrylate, and poly(styrenesu1fonate) which indicate the existence of charge fractions, t-l, at siple salt concentrations over 0.1 M. In the sae paper, Manning4 eployed a discrete linear array of charges to calculate the electrostatic portion of the total free energy of the polyelectrolyte syste. At finite salt concentration, Cs, such a odel ay not be appropriate for a and DNA helices, hence a partial otivation of the present approach. Moreover, Bailey6 has sued the Mayer cycle diagras and has concluded that assuptions I and I1 are accurate for Cs Recently, Stigter7 has published a coparison of nuerical solutions of the nonlinear Poisson-Boltzann equation with Manning's counterion condensation theory. His work supports to soe extent the results of MacGillivray and Winkleand and MacGillivray9Jo on the zero salt liit validity of Manning's theory. However, at finite salt and especially for E > 1, significant differences in the surface potential fro predictions of counterion condensation theory are apparent. Clearly, work reains to be done to reconcile the differences between the Poisson-Boltzann equation and counterion condensation approaches. As charge screening increases with an increase in supporting electrolyte concentration, the neglect of POlyion-polyion interactions, assuption 111, should becoe a better approxiation at finite salt concentration. In the case of cy helical and DNA-like helical polyelectrolytes, the infinite length approxiation iproves with increasing Cs. If the range of the electrostatic interaction is sall relative to the length of a rodlike region, the neglect of end effects is valid. As both cy helices and DNA have very large intrinsic persistence lengths, assuption IV is quite reasonable. When considering linear polyions (e.g., carboxyethylcellulose) that are approxiated as infinite pitch helices, care should be taken to insure the above criterion is et.l' We conclude this section with a brief exaination of assuption V. A priary purpose of this work is the investigation of the effect of a dielectric discontinuity on helical polyelectrolyte colligative properties. Hence, we have arbitrarily chosen the cylinder dielectric constant Dl as 2; the bulk solvent dielectric constant, D2, was taken to be 80. Moreover, we have placed the line of charge on the surface of the dielectric cylinder. Based on the work of Bailey,12 we observe that the actual location of the charge (in or on the cylinder) is uniportant in the deterination of FeXce,, for a and DNA-like helical distributions. However, in the case of the infinite pitch helix,13 whether the charge resides on or is ebedded in the cylinder is significant. As the line of charge is placed closer to the center of the low dielectric region, those ters which represent the angular asyetry in the potential becoe ~niportant.'~ Consequently, the proposed infinite pitch helical odel ay be appropriate for those polyelectrolytes with large backbone radii, e.g., carboxyethylcellulose, but is perhaps unrealistic for sall cylinder radii olecules such as poly(acry1ic acid). B. Calculation of FeXCeSS and Soe Colligative Properties. Let the jth charge in a helical array be located at z, in the cylindrical polar coordinates (z,,a,o') = r'. Here, 0' = 2nz,/p and the pitch of the helix is p. For an a helix, pia is about 0.68 and for a DNA-like helix, Pia is about When the pitch is infinite, the helix becoes a line of charges parallel to the cylinder axis. In paper 1 we solved the linearized Poisson-Boltzann equation for the potential of a point charge on the surface of a obile ion-free, low-dielectric cylinder. Whereupon, the potential \kt arising fro the interaction of two charges on the surface of the cylinder and a distance z apart15 (with respect to the principal axis) is given fro paper 1 (eq 2.11) by 2q \k - -{H, + 2 E H, cos n($ - e')) (la) - ZT H, = z l dlh, cos (k) 0 (1b) The I, and K, are odified Bessel functions of the first and second kind.' The prie denotes the derivative with respect to the arguent. Furtherore, a is the radius of the dielectric cylinder and X2 = l2 + K ~ where, K - ~ is the Debye screening length: n, nuber of ions of species i per c3 and u, is the valence of the ith ion. The potential \I, at a given charge in the helical array is defined as the su of the pairwise interactions \kt 9 = C*T(K,Zj) (2) J The sus could be handled straightforwardly with the aid of ethods presented in paper 1. It would be substantially easier if the sus could be converted to integrals; we shall odify eq 2 to perit this siplification. Before considering the continuu liit of eq 2, we note that corrections arising fro the discrete nature of the charge distribution are expected when Kb is on the order of unity. For a DNA helix, the discreteness of the distribution becoes iportant when Ka N 5.88 or Cs = Cs is the olar concentration of siple salt. These observations are confired by Bailey;12 Bailey considered a discrete charge representation of DNA and found for Cs up to 1.1 M (Ka = 3.3 and Kb = 0.6) only sall, salt-independent corrections to the uniforly charged cylinder surface potential. As we are interested in that part of \k which depends on salt concentration, the continuu liit of eq 2 should be a valid representation of the physics for Ka 5 2. As written, conversion to an integral would give the potential acting on a charge in a continuous helix, and this potential is infinite. We, therefore, add and subtract a coparison potential for vanishing salt concentration and write \k = A\k + \ko (3) = E\k'TO(O,z,) (4) J = C[*T(K,Z]) - 'kto(o,z~)l (5) J If \ko were defined as the potential \k evaluated at K = 0, the suand in eq 5 would be finite at z = 0 but would diverge for large z,, whereupon the desired integral over z could not be extended to infinity. Consequently, qo is defined as the potential for K = 0 inus the potential of a uniforly charged cylinder at K = 0. The subtraction is identical to an oission of the n = 0 ter in eq la (the n = 0 ter is proportional to -In a). We thus arrive at a

3 Vol. 12, No. 3, May-June 1979 Colligative Properties of Helical Polyelectrolytes 517 Table Ia 4nAhn /n , , , a Cn Ah, as a function of n and Ka for an 01 helix, pia = 0.68, D, = 2, D, = 80. definition of A\k which perits replaceent of the su by an infinite integral. As the reference potential, 'ko, is independent of salt concentration, A\k separates naturally into the potential of a unifor charge distribution and corrections that depend on the pitch. Eploying eq la in eq 4 and in the continuous liit, we find P + A+ = - C s 7T n=- - dls - dz(hn - hno) exp[i(lz + n4)l (64 Here P is the charge per unit length of axis, and 4 = 8 + 2az/p (6b) We have included a phase shift 8 to incorporate the effect of the double stripe of charges in DNA. If 8 = 0, the z = 0 ter should be oitted; if 8 = 180, all charges are included in the su. Recognizing a delta function in eq 6a, we have 6(y) = (27r-l Jl dz exp[zyl A'k = 2P(ho Ah,, cos ne) (74 Ah,, = h,,(2an/p) - hn0(2rn/p), n L 1 (7b) ho = Ko(KU)/D2KaKl(Ka) (7c) Here, the displayed arguent of h, is the value of 1. A superscript zero iplies K = 0, and its absence indicates that the actual value of K should be used. In the calculation of colligative properties, the quantity of interest is the difference in reversible work done in charging up a helical distribution of charge in the presence and absence of salt, Fercess, and is related to A'k by , E , , of radius a. The second class of ters contains corrections to the excess free energy due to differences in the helical charge distribution fro a unifor one. When the pitch p goes to zero, all 4hn = 0 and we recover the uniforly charged cylinder result. The colligative properties of the polyelectrolyte solution can be related to (af5xcess/a~)t,v2 The first ter, proportional to -1, arises fro the interaction of a line of charge iersed in bulk solvent with the obile ions. The second ter, proportional to K,2(KU)/K12(KU), contains corrections to (afexcess/a~)~,~, resulting fro the fact the helix is wrapped around a salt-excluding cylinder. For a fixed value of K, it is an increasing function of KU. Increasing KU gives rise to a larger excluded salt effect and a concoitant increase in FexceSs. The third class of ters are the corrections to F,,,, due to the difference in the helical distribution fro a unifor one. Included in d;'excess = 2P2L C Ah,, cos n8,,fexcess are the effects of obile ion screening. Hence, d;'ex,,ss should be a decreasing function of KU. In the cy helical case, a = 7.5 A, p = 5.1 A, and 8 = 0'. The required values of h, are easily coputed fro eq IC. Consultation of Table I readily verifies that to an excellent approxiation Substitution of eq 7a into eq 8 and integration over z gives Fexcess Fexcess = E( + 2D2 C Ahn cos n8 Dz KCZKi(KU) The first ter on the right-hand side of eq 9 is the excess electrostatic free energy of a uniforly charged cylinder and

4 518 Skolnick Table IIa p- t 2 ~ i~line - excess VS. Ka Ka unifor cylinder e = 0" o a For a helix of infinite pitch, D,, the solvent dielectric constant was taken to be 80. D,, the cylinder dielectric constant, was assued to be 2. Macroolecules o KA Figure 1. The triangles give the value of -Ca(Ka) as a function of KCL, if the a helix were a line of charge in bulk solvent. The stars give the value of -Ca(Ka) fro eq 12 and incorporate the salt excluding backbone effect into the colligative properties. p = 5.1 A and a = 7.5 A. I I 1 Thus, for a continuous cy helical distribution of charge, the excess electrostatic free energy is well approxiated by that of a uniforly charged cylinder. Moreover, it is only in the liit that Ka - 0 that the helical distribution appears as a line of charge. In Figure 1, we plot -C"(KU) as a function of Ka, where If KU or Cs I 3.7 X M, we predict experientally observable (greater than 5%) deviations in the colligative properties of cy helices fro a line of charge in the bulk solvent odel. If pitch is set equal to infinity, the helix reduces to a line of charge ebedded on the surface of a low-dielectric, salt-excluding cylinder. This odel ay be relevant to linear, nonhelical polyelectrolytes such as carboxyethylcellulose. Taking the liit of Ahn in eq 7b as the arguent of h, goes to zero, i.e., the pitch, p, becoes infinite, it follows that12 where Ah,line = o K A Figure 2. The stars give -Che(Ka), defined in eq 15, as a function of Ka for a line of charge ebedded on the surface of a low dielectric, D1 = 2, cylinder; -1, the triangles, is the value if the line of charge were iersed in bulk solvent. The circles denote the uniforly charged cylinder contribution to -CIine(~a). Table 111" a vs. Cs ax a, A Csax, ol/l a a vs. the axiu salt concentration line of charge in bulk solvent adequately (within 5%) predicts the colli. gative properties of a helix of infinite pitch. (aft;:ess/a~)t,v has been calculated. Taking the derivative of F!::ess with respect to K we have -p2l Cline( KU) (14) where As previously, the first ter on the right-hand side of eq 13a is the free energy associated with the reversible work done in charging up a uniforly charged cylinder. The second class of ters appears due to the angular asyetry in the lines of flux induced by the presence of the low-dielectric, obile, ion-free cylinder. In Table 11, we as a function of Ka; discussion of the results is deferred until after (15) In Figure 2 we plot -Cline(,u) as a function of Ka. As expected, when KU << 1, the doinant contribution to (df!;:ess/a~)t,v arises fro the line of charge in the bulk solvent odel. Provided that the cylinder's radius is very

5 Vol. 12, No. 3, May-June 1979 Table IVa unifor av of e + n, KO cylinder 0 = 0 e = n i.e., A ~DNA a For a DNA-like helix pia = 3.45, D, = 2, D, = 80. sall relative to K, the obile ions essentially interact with a line of charge with a slight perturbation caused by the dielectric cylinder. However, when KU , there should be easurable deviations in -Che(~u) fro a line of charge in bulk solvent odel. In Table 111, we present the axiu salt concentration vs. cylinder radius for which the line of charge odel in bulk solvent adequately characterizes the colligative properties of an infinite pitch helix. Note that -C*lne(~a) is a decreasing function of KU for a fixed value of K. This ay perhaps be rationalized by a qualitative argueent analogous to that for the two point charges (see the discussion at the close of section I1 in paper 1). First of all., it is those ters which reflect the angular asyetry in the potential, a( 5 Ah,line) 2D2( '=' a In K ) that doinate the difference in (af~~~ess/ak)t,v fro -1. (The line of charge in bulk solvent results.) As previously, we argue that the lines of flux will tend to avoid the cylinder, the avoidance increasing with increasing KU. Whereupon, over a region of space opposite the line of charge (0 = 180 ), there is a decreased flux density fro both the spreading of the lines of flux and their terination on counterions. Conversely, in that portion of space near (0 = 0') the line of charge, there is an increase in flux density vis-h-vis the absence of the dielectric cylinder. With respect to the line of charge in bulk solvent, the net effect appears to be a decrease in the excess reversible work required to charge up a helix of infinite pitch. Thus, the excess reversible work decreases with increasing angular asyetry, i.e., increasing KU. One further note is necessary: the qualitative conclusions of the infinite pitch helix result should hold for all D1 << D2; this calculation points out the qualitative iportance of the low dielectric effect on the colligative properties. We continue this section with an exaination of the colligative properties of a DNA-type double helix. The charge distribution is odeled as two helical lines of charge, 180 out of phase with respect to each other. The lines of charge lie on the surface of a low-dielectric, T, V Colligative Properties of Helical Polyelectrolytes 5 19 O' i I I I I o KA Figure 3. The circles give -CDNA(), defined by eq 18b, as a function of ~ a The. squares represent the uniforly charged cylinder contribution to -CDNA(); -1, the triangles, is the value for a line of charge in bulk solvent. p/a = salt-excluding cylinder. Thus, by eq 7a and by adding the potential of the two helical lines 4qDNA = 2P(h0 + 2CAhnDNA(1 + COS (n~))] (164 P is the charge density per unit length of axis; in DNA the unit length of the axis is 1.7 A. Clearly, only the even n contribute to the su in eq 16a. Hence, 4qDNA = 2P{h AhpnDNA] (16b) We have coputed the ah2, fro eq 7b. The corrections to the unifor charge distribution are extreely sall for KU < 1.0 as a brief consultation of Table IV will verify. Inserting eq 16b into eq 8 gives FFZ',A,, = "( + 4D22 AhZnDNA D2 KUK~(KU) Hence, where nu, - I I K12(~a) (18b) In Figure 3 we plot -CDNA(~a) as a function of KU. Whenever KU or Cs L 2 X M for DNA, appreciable corrections to the colligative properties of a line of charge odel are necessary. If KU I 0.5, we find a contribution to CDNA(~u) due to deviations in the double helical charge distribution fro cylindrical syetry. Such effects are to be expected when the screening length is of the order of the pitch or saller. Furtherore, corrections to cylindrical syetry ust also depend on the ratio of' the pitch to the cylindrical radius. That is, for fixed KP whether or not the charge distribution appears unifor depends on how tightly wound the helical lines are. In the case of DNA double helix, P/a is 3.45 and KP = 1 when KU = 0.3. Thus, corrections to the unifor charge result arise in CDNA(~a). On the other hand, an cr helix has a pitch to radius ratio of 0.68, and KP = 1 when KU = 1.5. It is therefore not surprising that for KU < 2 corrections

6 520 Skolnick Macroolecules to the unifor charge distribution are negligible in the a-helical case. This section is concluded with a brief presentation of soe corrected colligative properties for cy-helical and DNA-type double helical charge distributions. We shall eploy Manning s notation2 and assue only onovalent obile ions are present. Define f = $ {DzkBTb)- b is the linear spacing of the charges along the principal axis. When f < 1, the activity coefficients yi of the obile ions are2 Here 2 equal to 1 refers to counterion and 2 to coion; ni is the nuber density of obile ions of species i, and 6 = cy, line, or DNA refers to the cy helix, line of charge on the dielectric cylinder, and DNA double helical polyelectrolytes, respectively. Now, Consequently, we can write if f < I -f(x)(x + 2)-1 In yi = C6(KU) (19c) 2 Here C%a) is given by eq 12, Cline(,u) is given by eq 15, and CDNA(~a) is obtained fro eq 18b. X is the ratio of the concentration of counterions fro the polyelectrolyte, ne, to the concentration of counterions fro the siple salt, n,. The ean activity coefficient y*6 is: 1n Y*6 = Y *b Y1 + ln Yzl (204 The osotic coefficient $* is related to the ean activity coefficient in the Deybe-Huckel approxiation by2 Eploying eq 20b, we have $* = 1 + In yi* 4 < 1 (214 fx(x + 2)- (21b) 2 In the liit that ns - 0, X - and C*(KU) Denoting the salt-free osotic coefficient by $p6, we have $p6 = 1 - f/2 f < 1 (22) in agreeent with Manning. For f > 1, we proceed in an identical fashion as Manning does to find ( -lx ex.[ -E- XC (K U)] + f >1 (23a) y16 = (X + 1) 2(f- X + 2) 4xn2 (244 $p6 = (2f)- f > 1 (24b) A word of explanation with regards to eq 23a through eq 24b is appropriate. One of the referees has correctly observed that at high salt, e\k/kbt becoes sall and for f > 1 the Poisson-Boltzann equation can be linearized; i.e., condensation ust actually disappear if the Poisson-Boltzann description is correct. On the other hand, there is both theoretical12 and experiental evidence4 which indicates Manning theory predictions for the condensed charge fraction ay be correct to at least 1.0 M. Thus, we regard eq 23a through eq 24b as a logical theoretical extension of Manning s original work; the equations deonstrate, within the context of a counterion condensation odel, the influence of finite cylinder size and dielectric discontinuity on polyelectrolyte colligative properties. By analogy to siple electrolyte theory, it sees appropriate to designate equations 19c through 24b as extended Manning colligative properties. It follows fro eq 21b, 22, and 24a,b that the so-called additivity + 272,) = 4p6ne + 2n, (25) holds only in the liit that KU - 0, i.e., C*(KU) - 1. This conclusion casts further doubt on the interpretation of eq 25 which states that a fraction (1- &) of the counterions fro the polyelectrolyte salt are bound to the polyion. Strictly speaking eq 25 is valid for helical polyelectrolytes when they appear as a line of charge, in other words at infinite dilution. However, as a atter of practical application, eq 25 is a useful approxiation whenever KU I Additional Observations We conclude our discussion with several observations on Manning s counterion condensation theory. In his investigations, Manning replaces the actual flexible, linear polyelectrolyte by an infinitely thin line of charge. To the uninitiated and perhaps naive, ignoring the influence of the cylinder on the excess electrostatic free energy sees to be an approxiation of questionable validity. Such a query provided part of the original otivation for this work. We therefore exained the colligative properties of helical polyelectrolytes and found when Ka L 0.15 a line of charge odel is inadequate. On the other hand, when the screening length is large relative to a, the helical polyion appears as a line of charge and Manning s original treatent is appropriate. Consequently, we forulated a series of extended Manning colligative properties for all values of Ka I 2. Acknowledgent. The author would like to thank Marshall Fixan for invaluable assistance and stiulating discussions throughout the course of this work. This work was supported in part by N.I.H. Grant GM References and Notes (1) J. Skolnick and M. Fixan, Macroolecules, 11, 867 (1978). (2) G. Manning, J. Che. Phys., 51, 924, 3249 (1969). (3) G. Manning and A. Holtzer, J. Phys. Che., 77, 2206 (1973). (4) G. Manning, Biophys. Che., 7, 95 (1977). (5) H. S. Kielan, J. M. A. M. van der Hoeven, and J. C. Leyte, Biophys. Che., 4, 103 (1976), and references therein. (6) J. M. Bailey, Biopolyers, 12, 1705 (1973).

7 Vol. 12, No. 3, May-,June 1979 Viscosity of Branched PDMS 521 (7) D. Stigter, J. Ph3.s. Chern., 82, 1602 (1978). (8) A. D. MacGillivray and 3. J. Winkleann, J. Che. Phys., 45, 2184 (1966). (9) A. D. MacGillivray, J. Che. Phys., 56, 80, 83 (1972). (10) A. D. MacGillivray, J. Chern. Phys., 57, 4071, 4075 (1972). (11) J. Skolnick, Ph.D. Thesis, Yale University, (12) J. M. Bailey, Biopolyers, 12, 559 (1973). (13) We are indebted to referee nuber two for pointing out the liitations of the infinite pitch helix odel. (14) D. Soupasis, J. (:he. Phys., 69, 3190 (1978). (15) We are grateful to a referee for supplying an additional reference to F. E. Karasz and T. L. Hill, Arch. Bioche. Biophys., 97, 505 (1962). Karasz and Hill derive the interaction between two point charges located inside a salt-excluding dielectric cylinder which has been placed in bulk salt solution. 0, given in eq la, is a liiting case of their treatent. As indicated previously,'j1 when the point charges are brought fro within to on the dielectric cylinder, the contribution of those ters which reflect the distortion of the lines of flux due to a dielectric discontinuity becoes iportant. Structure and Viscosity of Poly(diethylsi1oxanes) with Rando Branches Enrique M. Valles and Christopher W. Macosko* Departent of Cheical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota Received May 17, 1978 ABSTRACT: Viscosity rise and extent of reaction were followed during the stepwise polyerization of vinyl-terinated poly(diethylsi1oxane) with tri- and tetrafunctional hydrosilanes. Gel point results and M, data by light scattering agree with predictions fro branching theory. This indicates that the influence of side reactions, substitution effects, and ring foration is sall. The bulk viscosity of the branched olecules was found to correlate with the weight average olecular weight of the longest linear chain through the olecules, ML,w, and with gm,, where g is the ratio of hranched to linear polyer radii of gyration. A ajor area of study of the physics of polyer science is the one concerned with the relations between easurable viscoelastic properties and olecular paraeters. One of the best known relations in this area is the one that correlates the zero shear rate viscosity of polyer elts to their average olecular weight. If the polyer is linear, 90 = KM," with a = 3.4 for M, > M, and a for M, < M,, where K is a teperature-dependent constant, qo is the zero shear rate viscosity, M, is the weight averagi, = olecular weight, and M, is the critical olecular weight beyond which the influence of entangleents begins to be iportant. Equation 1 applies to all coon polyers, both in bulk and in concentrated solution.' For branched systes the relation between qo and the olecular paraeters is not so well established. The introduction of a few long branches sees to decrease the viscosity of the polyer if it is copared with the linear polyer of the sae olecular weight.2-7 However, when the branches are long enough to for any entangleents, the viscosity of the branched syste is higher than that of the linear polyer of the sae olecular Graessley" has recently reviewed the effect of long branches on qo for odel branched aterials, four- and six-ar star olecules. He finds a good correlation fix the viscosity data with where a = 3.4, the sae as for the linear case, and g is the ratio of the squares of the radii of gyration of a branched to the linear chain of the sae M,, sb2/s?. Thus g I 1 and decreases with increased branching. Graessley finds that eq 2 holds up to gkl, = 2 x lo5. At higher gm, values, presuably when thle branches entangle, qo rises even ore steeply for these branched polyers as has been observed in other systes.s-'o Randoly branched olecules, rather than regular stars, /79/ $01.00/0 C Table I Reactants B, ViPhCH,SiO(Si(CH,),O), ' 2.21 B,' SiCH,PhVi ViPhCH,SiO(Si(CH, j20)n- SiCH,PhVi 33400' 2.60 A, (HSi(CH,),O),Si 329 A, (HSi(CH,),O),SiPh 330 A, H(CH,),SiOSi(CH,),H 134 ' By vinyl titration (ref 14); average of six titrations ranging fro to ' Sae as a with three titrations ranging fro to Reactant identification. are the type which occur in ost coercial polyerizations. However, their structure, and thus g, is difficult to deterine. By step-reaction polyerization a randoly branched syste of known structure can be obtained using a polyfunctional coonoer. In this work we have studied the viscosity rise during polyerization of a well-defined syste coposed of long olecules of bifunctional poly(diethylsi1oxane) with reactive groups at their ends and sall trifunctional and tetrafunctional olecules. First we describe this polyerization syste and our experiental ethods. Then we use the branching theory to relate the viscosity rise data to the changing structure of the reacting syste. Experiental Section The Cheical Syste. Polyerization between long olecules of end-vinyl-substituted poly(diethylsi1oxane) and silanes of three different functionalities were studied. A coplete listing of the cheicals used, together with their average olecular weights and olecular weight distributions, is given in Table I. As an exaple, the reaction between the vinyl-terinated Aerican Cheical Society