J. Phys. Chem. 1992,96, 3915-3922 98. Walter Hugo Stockmayer 70. R. Koningsveld and M. Fixman, Macromolecules 17, 507-508 (1984). 99. Non-Gaussian Contribution to the Hydrodynamic Radius a t the Theta Point. M. Fixman and M. Mansfield, Macromolecules 17, 522-527 (1984). 100. Failure of Universality for Translational Diffusion of Chain Polymers. M. Fixman, J. Chem. Phys. 80, 6324-6325 (1984). 101. Absorption by Static Traps: Initial-value and Steady-state Problems. M. Fixman, J. Chem. Phys. 81, 3666-3677 (1984). 102. Virial Coefficients for Diffusion, Conductivity, and Dielectrics. M. Fixman, J. Phys. Chem. 88, 6472-6479 (1984). 103. Entanglements of Semi-dilute Polymer Rods. M. Fixman, Phys. Rev. Lett. 54, 337-339 (1985). 104. Dynamics of Semidilute Polymer Rods: An Alternative to Cages. M. Fixman, Phys. Rev. Lett. 55, 2429-2432 (1985). 105. Translational Diffusion of Chain Polymers. I. Improved Variational Bounds. M. Fixman, J. Chem. Phys. 84, 4080-4084 (1986). 106. Translational Diffusion of Chain Polymers. 11. Effect of Internal Friction. M. Fixman, J. Chem. Phys. 84, 4085-4090 (1986). 107. Dynamical Problems in the Theory of Polymer Solutions. (Plenary lecture IUPAC Symposium on Macromolecules, The Hague, August, 1985). M. Fixman, Makromol.
3915
Chem., Macromol. Symp. 1, 17-22 (1986). 108. Implicit Algorithm for Brownian Dynamics of Polymers. M. Fixman, Macromolecules 19, 1195-1204 (1986). 109. Construction of Langevin Forces in the Simulation of Hydrodynamic Interaction. M. Fixman, Macromolecules 19, 1204-1207 (1986). 110. Brownian Dynamics of Chain Polymers. M. Fixman, Faraday Discuss. No. 83, 199-21 1 (1987). 111. The Shear Melting of Colloidal Crystals: A Long Wavelength Driven Transition. P. Harrowell and M. Fixman, J. Chem. Phys. 87, 4154-4161 (1987). 112. Dynamics of Stiff Polymer Chains. M. Fixman, J . Chem. Phys. 89, 2442-2462 (1988). 113. Chain Entanglements. I. Theory. M. Fixman, J. Chem. Phys. 89, 3892-3911 (1988). 114. Chain Entanglements. 11. Numerical. M. Fixman, J. Chem. Phys. 89, 3912-3918 (1988). 115. Theory of Polymer Dynamics. M. Fixman, In Frontiers of Macromolecular Science, Proceedings of the 32nd International Symposium on Macromolecules, MACRO88, Kyoto, 1988; T. Saegusa, T. Higashimura, and A. Abe, Eds.; Blackwell: Oxford, 1989. 116. Models of Polymer and Solvent Dynamics. M. Fixman, J . Chem. Phys. 92, 6858-6866 (1990). 117. Polyelectrolyte Bead Model. I. Equilibrium. M. Fixman, J . Chem. Phys. 92,6283-6293 (1990). 118. Stress Relaxation Theory. M. Fixman, J. Chem. Phys. 95, 1410 (1991).
Acid-Base-Induced Association of Amino-Terminated Polystyrenes. 1. Linear Chains and Ring Formationt Gerhard Merkle and Walther Burchard* Institute for Macromolecular Chemistry, University of Freiburg, Stefan- Meier-Strasse 31, D- 7800 Freiburg, Germany (Received: August 28, 1991)
The associating coupling process of tertiary-amino-terminated polystyrene (FS) chains with monofunctional and bifunctional dinitrophenol reagents has been studied by means of UV-vis spectroscopy and light-scattering measurements as a function of polymer concentration. The mechanism of coupling was first checked with monofunctional PS chains. The equilibrium constants of the two functional groups of the coupler were found to be slightly different from each other; but with the same average value as for the monofunctional phenol compound. The increase of molecular weight agreed well with predictions from Flory's theory. Much smaller weight average molecular weights were found, however, for the coupling of telechelic PS chains, and the deviationsare stronger for the short chains than for the longer ones. The effect is explained by ring formation and three approximationswere applied. These are (a) a mean field approximation, (b) the Jacobson-Stockmayer (JS) approach, and (c) the JS approach including excluded volume effect and its influence on Gaussian statistics. Molecular weights were also calculated from the weight fraction of rings and fitted to those obtained by LS where the effective bond length b was chosen to be the adjustable parameter. In the original JS treatment the effective bond length increased with the chain length of the primary chain. In treatment c this could be eliminated by consideration of excluded volume, but absolute values of b are unrealistically low.
1. Introduction Recently we started a study of well-defined models to achieve a better understanding of some essential principles in physcial gels.',2 Our idea was to prepare star-branched macromolecules, which alone have no tendency of self-assembling, but which can be coupled by a bifunctional compound via the terminal ends of the arms. Such a system was first studied by W o r s f ~ l d . ~We .~ took up this problem again and are trying to deepen this study by a thorough use of new experimental techniques in combination with current theories. Dedicated with great respect to the 65th birthday of Professor Marshall Fixman.
0022-365419212096-3915$03.00/0
This system represents exactly the well-known and famous random branching model that was treated in 1943 by StockmayerS who predicted the weight average molar mass from the extent of reaction. The functionalities of both components are known, and the junctions are point like but reversible according to an acid-base (1) Te Nijenhuis, K.In Physical Nerworks, Polymers and Gels; Burchard, W., Ross-Murphy, S.B., Eds.; Elsevier Applied Science: London, 1990 pp 15-33. (2).Burchard, W.; Stadler, R.; Freitas, L. L.; Moller, M.; Omeis, J.; Mhhleisen, E. In Biological and Synthetic Networks; Kramer 0.. Ed.; Elsevier Applied Science: London, 1988; pp 3-38. (3) Worsfold, D. J. J. Polym. Sci., Polym. Phys. Ed. 1982, 20, 99. (4) Worsfold, D. J. J. Polym. Sci., Polym. Chem. Ed. 1983, 21, 2271. (5) Stockmayer, W . H. J. Chem. Phys. 1943, 1 1 , 45.
0 1992 American Chemical Society
3916 The Journal of Physical Chemistry, Vol. 96, No. IO, 1992
mechani~m.?~Coupling is caused by a dinitrophenol compound, and the extent of reaction and equilibrium constant can be determined by UV spectroscopic techniques. Concentration dependence of molecular weights and z-average diffusion coefficient 0,can be measured by combined static and dynamic light scattering (LS).6 Thus a rigorous test of the validity of the Flory-Stockmayer’ approach is possible. Polymer scientists often successfully apply Stockmayer’s theory to permanent networks in which cross-links are irreversibly formed. It is mostly forgotten that Stockmayer did not explicitly treat such systems. His method is equivalent to assuming chemical equilibrium, this is to say, bonds which are formed and broken reversibly. For our study we used the same coupling reagent as Worsfold, bis(2,6-dinitrohydroquinone)ester of adipic acid (I), but increased the functionality of the polymers from mono- to bifunctional (telechelic) polymers and trifunctional star molecules. This paper only deals with the static properties of linear products, since their treatment yields the indispensible fundamentals for association of star molecules. The branched products, caused by association of trifunctional polymers and the dynamic properties (viscosities and diffusion coefficients), will be presented in later papers.
Merkle and Burchard TABLE I: Molecular Weights, Polydispersity, and Seeond Virial Coefficients in (mol mL/e*) X lo‘ from Amino-Terminated PS sample MJLS) MNo MNb MwIMNc AZC A i M01 5000 4800 1.07 12.2 MO2 10000 9000 1.03 8.8 8.9 6.5 M03 3200 3200 3000 1.05 BIl 17500 13800 14800 1.3 8.2 (3.9) 7.8 B12 35000 30500 30300 1.3 6.6 5.1 5.5 1.2 6.1 4.55 4.80 B13 50000 47000 50000
From osmosis measurements. From end group analysis by UV-vis solvent toluene. ‘In dioxane. ’In spectroscopy. ‘From SEC. dioxane with an equimolar amount of monofunctional phenol.
ino-terminated polystyrene became visible by treating the silica gel plates with iodine vapor. With toluene as eluting solvent only the nonfunctionalized polymer could be eluted, while THF eluted both polymers. The amino-terminated polystyrene could be readily separated in large scale from nonfunctionalized polystyrene by silica gel (mesh 70-230) chromatography with toluene as eluent until all of the polystyrene had eluted, followed by THF to elute the amino-terminated PS. 23. Synthesis of Dinitropbeaols. The adipyl ester was prepared by the method of Worsfold3 from adipoyl chloride and 2.6-di2. Experimental Section nitrohydroquinone. The latter was obtained from reaction of 2.1. Synthesis of Polymers. Polymers were prepared by hydroquinone with acetic anhydride,I4 nitration with HN03,1S,16 standard anionic polymerization technique under a slight argon and saponification by methanolysis.” excess Tertiary amino groups were introduced either 2.4. Size Exclusion Chromatography (SEC). SEC measureby initiating polymerization of styrene with 3-(dimethylamino)ments were carried out on a chromatograph fitted with refractive propyllithiumIO(DMAPLi), or by terminating living carbions with index and UV detectors. Four columns of porosities lo6,lo5,104, 3-(dimethylamino)propyl chloride (DMAPCI) .I Monofunctional and lo3 A and solvent chloroform were used. polymers, in the following denoted by MO, were prepared by both MJM, was found to be 1.1 and lower for the monofunctional methods in solvent benzene, while bifunctional polymers, desigpolymers, and about 1.3 for the bifunctional polymers prepared nated by BI, were prepared by initiating the polymerization with in THF (Table I). potassium naphthalide in tetrahydrofuran (THF) at -78 OC and 2.5. Vapor Pressure Osmometry (VPO). VPO measurements terminating the reaction with DMAPCl. The amine chloride was (for sample M 0 3 ) were carried out on a Hewlett-Packard 115 isolated and purified as previously described.”,I2 Before use for molecular weight apparatus. The instrument was calibrated with the termination reaction, a benzene solution (10%) was once again bibenzoyl in chloroform at 30 OC. dried over CaH,, degassed on a high-vacuum line and distilled 2.6. Membrane Osmometry (MO). M O measurements were into a graduated dropping funnel. The initiator DMAPLi was made in THF on a Hewlett-Packard 502 high-speed membrane prepared according to the method of Eisenbach et a1.I0 in T H F osmometer equipped with a regenerated cellulose membrane. solution at -35 OC and immediately used after filtration. Po2.7. Preparation of Solutiolrs for IS and UV-Vi Spectroscopy. tassium naphthalide was made from naphthalene and potassium Stock solutions of about 5% polymer concentration were made in THF solution at room temperature. The initiator concentration up with highly purified dioxane, containing the equivalent amount was determined by the classical acetanilide method. of phenol group. About 15 dilute solutions were prepared (lowest The polymer solutions were filtered and recovered by precipconcentration = 0.05%) and filtered through 0.2-clm FTFE filters itation with methanol. In order to obtain reproducible light into dust-free cylindrical quartz cells for LS measurements. The scattering measurements it was necessary to centrifuge a 20% same solutions were used to determine the extent of reacted solution of polymer in dry benzene at 20000 rad/& for 2 h. The functional groups by measuring the absorption at the peak clear supernatant solution was decanted from a white precipitate. maximum of the phenolamine complex. The total concentration Analysis revealed that the precipitate consisted of salt and polymer. of phenol could be checked by the absorption at the isosbestic 2.2. Separation of Amino-Terminated and N ~ n f u n ~ t i ~ ~ l i ~ epoint.3 d PS. In contrast to termination with DMAPCl in THF, this 2.8. UV-Vis Spectrosc~py. Measurements were carried out reaction appeared to be very slow in benzene and took several on a thermostated PerMElmer W-vis spectrophotometer Model hours. The degree of functionalization thus obtained was only 330. For dilute solutions 1 cm2cells and for higher concentrations about 70%. The monofunctional polymers could be separated by cells of 0.1 cm path length were used. a chromatographic method, similar to that of Quirk et al.,I3 who 1,4-Dioxane (Uvasol, Merck) was further purified over an used it for separation of primary poly(styry1)amine. Thin layer alumina column (mesh 70-230, basic, activity I, Merck) directly chromatography (TLC) analysis were carried out on silica gel before use. Model compound N,N-dimethyldodecylaminewas plates G607254 (Merck), precoated with a fluorescent indicator, distilled from CaH2, and 2,6-dinitrohydroquinone-4-acetate,obwhich allowed detection of polystyrene with UV light. The amtained after nitration of hydroquinonediacetate, was twice recrystallized from toluene. The amino group content was measured spectroscopically by complexing with a large excess (30-fold) of (6) Bantle, S.; Schmidt, M.; Burchard, W. Macromolecules 1982, 15, 4-acetoxy-2.6-dinitrophenolin toluene. 1604. (7) Flory, P. J. Principles of Polymer Chemistry; Cornel1 University Press: 2.9. Determination of Equilibrium Constants. Stock solutions Ithaca, NY, 1953. of amine and phenol were mixed to yield about 10 solutions of (8) Hild, G.; Strazielle, C.; Rempp, P. Eur. Polym. J . 1983, 19, 721. different ratios 0.5 < r < 3, where r is the ratio of functional (9) Hild, G.; Kohler, A.; Rempp, P. Eur. Polym. J . 1980, 16, 525. (10) Eisenbach, C. D.; Schnecko, H.; Kern, W. Eur. Polym. J . 1975, I I , 699. ( 1 1) Broze, G.; Lefcbvre, P. M.; Jtr6me, R. Mukromol. Chem. 1977, 178, 3171. (12) Richards, D. H.; Service, D. M.; Stewart, M. J. Br. Polym. J . 1984, 16, 117. (13) Quirk, R. P.; Cheng, P. L. Mucromolecules 1986, 19, 1291.
(14). Horning, E. C., Ed. Organic Synthesis, Wiley: New York, 1955; Collective Volume 3, p 452. ( 1 5 ) Kehrmann, F.;Klopfenstein, W. Hela Chim. Acta 1923, 19, 1291. (16) Zemplen, G.; Schawartz. J. Acfu Chim. Acfo Sci. Hung. 1953,3,487. (17) Burger, A.; Fitchett, G. T. J . Am. Chem. SOC.1952, 75, 1359.
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 3917
Amino-Terminated Polystyrenes
0.14
'
2.00
Here, f is the functionality of the polymer and Mpand M,are the mofecular weights of polymer and coupler, respectively. Since the extent of reacted amino groups CY depends on initial concentration according to the acid base equilibrium it is possible to predict M, for any given polymer concentration when the equilibrium constants are known. 3.2. EqdibriumConstants. The equilibrium constant is defined as
I
2.50
3.00
3.50
4.00
4.50
* l o 6 I"[ 1. Refractive index increments dn/dc at 20 OC in dioxane vs 1/X2. The symbols denote sample ( 0 )M o l , (A) BI1, (V)BI2, and (m) M o l , (A), BI1, (v)B12 with coupler r = 1; (e) BIZ with coupler r = 0.5. l d
groups,and with total concentrations of functional groups ranging from 1.2 x IO4 to 4 X lo4 mol/L. Equilibrium constants were determined from the peak maxima of the phenol salt.3 210. Light Scattering (IS).The light-scattering measurements were carried out with an apparatus6 that allows simultaneous recording of SLS and DLS.6 A Model 2020 krypton laser (Spectra Physics) was used as light source (X = 647.1 nm) for the samples containing phenol compounds, and sometimes a Model 2020 argon laser (A = 488 or X = 496.5 nm) was used for pure polystyrene solutions. The dn/& measurements were performed with a BricePhoenix differential refractometer at three different wavelengths for each sample in dioxane. The data for solutions in cyclohexane or toluene were taken from the literature. Refractive index increments dn/& were determined in dioxane solution for polystyrenes (PS) and PS with coupling reagent of ratios r = 1 and r = to assess the influence of the dye on scattering behavior. The low molecular weight samples exhibited a molecular weight dependence of dnldc. When r = 1 these values are higher than for mixtures with r = or pure polystyrenes (Figure 1). 211. Estimrtionof ExperimenQl Errws. For the low molecular weights the accuracy of light-scattering measurement was about 10% and reduced to i5% for the higher molecular weights. The error in A2 is relatively high and may be estimated as 20%. The evaluation of A2 was based on 10 different concentrations in the range up to 30 mg/mL. 3. Theory 3.1. Weight Average Molar Mass M,. The quantity of particular interest is the weight average molecular weight as a function of polymer concentration. For known extent of reaction a,the weight average molar mass of the coupled PS chain can be calculated from Flory's treatment7 of linear cocondensation, under the assumption that no ring formation takes place. The extent of reacted functional groups is defined as follows: = [Al,/[Alo
(1)
B = [Blr/[B10
(2)
f f
[A], and [B], represent the concentrations of reacted functional groups and [AIo and [B], the total concentrations of amino and phenol groups, respectively. In all the experiments the ratio r was chosen to be r = 1. Then the molecular weight M, of a cocondensate consisting of two structural units different in molecular weight is given byl8 Mw
= (TI + T2) / T3
with
(18) Stockmayer, W. H. J. Polym. Sci. 1952, 9,69; 1953, 1 1 , 424.
(4)
[A] and [B] are the equilibrium concentrations of amino and hydroxyl groups, respectively. For the special case of equivalent numbers of functional groups, r = 1 and eq 6 reduces to
K=
--
a [AIo(l -
ffMP
(1 - ff)2fpc
(7)
with the polymer concentration
In previous studies by W0rsfold,3*~ the two hydroxyl groups of the coupler were shown to be different in reactivity, and two equilibrium constants kl and k2, according to reaction of first and m n d hydroxyl groups, have to be considered. The concentration dependence of a,for a step reaction with two equilibrium constants, can be evaluated from B j e r r u m ' ~ ' ~treatment, .~~ and the extent of reaction for r = 1 is given by a=
[CAI + 2[CA21 [AI r 2 W I + [CAI + [CA21) = 2[clo
(9)
[C] is the equilibrium coupler concentration, [CAI the concentration of once bound, and [CAI] the concentration of twice bound coupling reagent C; the total concentration IC], = 1/2[B]o. It is convenient to express the equilibrium constants in units of moles of reacted functional groups per liter, rather than in moles of the reactant, as is cu~tomary.~ The two equilibrium constants are thus defined as follows:
From (9)-( 1l), and with [A] = [AIo(1 - a),a cubic equation is obtained
with x = (1 - a). Only one of the three roots fulfills the condition 0 I a I 1. 4. Results 4.1. Dilute Solution Behavior of Amino-Terminated PS. LS measurements were performed in 1,4-dioxane, toluene, and cyclohexane to compare behavior with nonfunctionalized PS. In toluene A2 was, within experimental error, the same as for nonfunctionalized PS (Figure 2). The nonfunctionalized PS corresponding to M o l , measured in cyclohexane at 34.5 OC, gave an A2value greater than zero (A2 1.7 X 10-4 mol mL/g2), while the functionalized polymers M 0 1 and M 0 3 , on the other hand, appeared to be under i? conditions, Le., A2 = 0; there was no evidence for self-assembling. Hence, a direct polymer-polymer association may be neglected.
-
(19) Bjer", J. Metal Ammine Formation in Aqueous Solution; P. Haase and Son: Copenhagen, 1941. (20) Tanford, C. Physical Chemistry of Macromolecules; Wiley: New
York, 1961.
3918
The Journal of Physical Chemistry, VoI. 96, No. 10, 1992
-
10-2
N
. -cn
' 30
i
t
7
1
080
P--m
E
Merkle and Burchard
I I
360
d
I
cco 023
1
/ ~
0 03 12-6
M ,
,3
.2-2
,s-3
1
6
.e->
"
[g/moIl
Figure 2. A2 vs M, in toluene for PS chains: (0) Huber et a1.,2' (0) Bantle et a].: (A)Cowie et a].:' ( 0 )Berry,Mand (0)amino-terminated
PS. TABLE II: (A) Extinction Coefficients from Peak Maxima and Isosbestic Point in the Solvent Dioxane at T = 20 O C and (B) Equilibrium Constants, Association Enthalpy, and Entropy from Model Compounds
ccnc
f u r c i 2nd' Q - ~ L P S [rrc ' I ]
Figure 3. Extent of reaction a vs concentration of functional group: (0)
association of monofunctional model compounds; (A)coupler and N,Ndimethyldodecylamine; (V)coupler and M 0 2 . The continuous curve is a theoretical line calculated from eq 7 with one equilibrium constant K1 = 30000 L/mol; the dashed one is calculated from eq 12 with two equilibrium constants k , = 45 000 and k2 = 20 000 L/mol, respectively.
A
phenol A. nm e 356
4300
T,O C K,L/mol X AH,kJ/mol AS, J/(mol K)
complex X. nm e 8000
446
15 4.00
B 20 3.02
25 2.08 -42 -55
isosbestic point A. nm e 286
* 05 * 10
2050 30 1.49
40 0.80
To test the influence of end groups on A2, which result from the association of amino groups with phenol, we performed LS measurements in 1,Cdioxane on 1:1 mixtures of functionalized PS with monofunctional acetic acid ester of dinitrohydroquinone and observed no difference. 4.2. Equilibrium Constants. The absorption maxima of free phenol and the amine-phenol salt, the isosbestic point, and the extinction coefficients t are listed in Table 11. The data for the acetic acid ester and for the bidentate adipic acid ester agree with each other. The equilibrium constant K and its temperature dependence were determined in solvent dioxane, first with monofunctional model compounds, i.e., 4-acetoxy-2,6-dinitrohydroquinoneand N,N-dimethyldodecylamine.From the temperature dependence, association enthalpy and entropy were determined. The values are listed in Table 11. As expected from Wor~fold's~-~ experiment, we observed no difference in the value of the equilibrium constant when replacing the model amine by a polymeric chain amine. The concentration dependence of a at T = 20 OC was now calculated from eq 7 with the experimental value, K = 3.0 X lo4 L/mol, and is plotted in Figure 3 together with the experimental values of a obtained from UV-vis spectroscopic measurements of equimolar mixtures of model compounds. The excellent agreement between calculated and experimental values and the appearance of the isosbestic point confirm the assumption of a 1:l complex between amine and phenol in dioxane. Next, the difference in reactivity of the two hydroxyl groups in the bifunctional coupler had to be examined. To this end, we again measured the concentration dependence of a with 1:l mixtures (concerning functional groups) of the coupling reagent and model amine. The result is plotted in Figure 3 and shows a slight deviation from the pure statistical case; i.e., the two groups are not equally reactive. A direct determination of both constants is possible if k l / k 2is very large, such that the process takes place in two consecutive steps. Clearly, the reaction does not proceed in two such separate steps. We tried to estimate k2 under conditions where all first hydroxyl groups could be assumed to have reacted, Le. [B], >> [AIo. The equilibrium constant thus obtained was about k2 = 2.0 X IO4 L/mol. We then varied kl in eq 12 until the calculated values agreed with experimental data. A good fit was found with k, = 4.5 X 104 and k2 = 2.0 X 104 L/mol. The average equilibrium constant (klk2)'I2= 3.0 X lo4 L/mol and
S
I
0
C
30
90
60
c
120
750
180
[Q'I
F i e 4. (Kc/&)'12 against polymer concentration c in g/L of (0)M01 in dioxane and ( 0 )M 0 1 with coupler.
equals the value found for the monofunctional case. Another question is whether and how much the ratio k l / k 2 changes when the amino group is bound to a polymer chain. Worsfold3p4found a decrease of k2 of about 10%with increasing molecular weight from 12 X lo3 to 180 X lo3. This effect of polymeric chain length dependence is too weak to be observable by our experimental techniques; for the low molecular weight PS of this study the concentration behavior of a was found to be the same as for the model amine (Figure 3). Lastly, we had to examine the reactivity of the two amino groups in the telechelic polymers. The equilibrium constants, measured from solutions containing polymer B12 and a monofunctional phenol, were found to be k = 3.0 X lo4 L/mol, independent of ratio r and concentrations of functional groups. Thus, in contast to the coupler, the two functional groups in the telechelics are equally reactive. 4.3. Coupling of Monofunctional Polymers. Doubling of molecular weight is expected to be the only possibility of this reaction. Two polymers M 0 1 and M 0 2 (of 5 X lo3 and 10 X IO3 in molecular weight) were coupled with the bidentate dinitrophenol derivative (I). The increase of a with concentration has already been mentioned, and for both polymers the same result was obtained. (For clarity only one of them is plotted in Figure 3.) To investigate the concentration dependence of molecular weights, light-scattering measurements were performed from the same solutions. The low molecular weight samples induce no angular dissymmetry of scattered light. The commonly used light-scattering equation of Debye for small particles is given by Kc/RB = (l/M,,,)
+ 2 A 2 ~+ 3A32 + ...
I
1/MaPp(c)
(13)
All quantities have their usual meaning. The square root reciprocal apparent molecular weights M,,(c) of M 0 1 and of M 0 1 with coupling reagent are shown in Figure 4. As expected, monofunctional PS with coupler shows a higher molecular weight than the polymer alone when the data is extrapolated to zero concentration. At low concentrations dissociation of the dimeric complex into single chains is observed. The apparent molecular weights now have somehow to be corrected with respect to A2 and A 3 to obtain the real molecular weight M J c ) corresponding to each concentration. The second virial
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 3919
Amino-Terminated Polystyrenes
,
2.20
9
f-
2
2.00
-
1.80
-
I
1
1.40 1.60
t
1.20
-
0.20
1.00
d
000
10-6
10-3
10-5
10-2
io-’
,0-5
c [mol/Il
Figure 5. M,(c)/M,(O) of coupled monofunctional PS against polymer M o l , (A) MO2. The full line is calculated from eq concentration: (0) 4 with& = I , f , = 2, kl = 45000 L/mol, k2 = 20000 L/mol, M, = 520, and Mp= 5000 or Mp = 10000.
coefficients of PS were measured in dioxane, but in an associating system we may postulate21,22
-
I
/
/ conc
’0.4
ic-3
fwclional grcups
10-1
q0-2
‘POI/
]
Figure 6. Extent of reaction a against concentration of functional groups for coupling of telechelic PS: ( 0 ) BIl, (A) B12, (V)BI3. The continuous curve is calculated from eq 15 with k, = 45 000 and k2 = 20 000 L/mol, respectively.
- 1
I
Rg(c)3/Mw(c)2 (14) with R, = ( S 2 ) 1 1 /and 2 , A2 is a function of M,. A power law is observed for irreversibly bound chain^,^^,^^ and similar behavior can be assumed for reversibly bound chains, if the molar mass is large enough. This yields A2
A2(C) = KAMw(C)-O (15) Here A2(c)is the second virial coefficient which the particles of molar mass M,(c) would have at a finite concentration c. It is not directly measurable but is assumed to be related to Mw(c) according to eqs 14 and 15. The constant factor KA may be eliminated23by considering the limit
0.00
020
040
060
080
100
a
F w 7. M,(c)/M,(O)
against a for coupling of telechelic PS: (0) BI1, (A) BI2, (V),BI3. The continuous curve is calculated from eq 4 with fP = fc = 2 and the corresponding molecular weights.
from eqs 4 and 12, but now CY is expressed in terms of polymer concentration using k, = 4.5 X lo4 and k2 = 2.0 X lo4 L/mol. 4.4. Coupling of Bifunctional Polymers. Again the ratio of amino groups to hydroxyl groups was chosen to be 1:1. Chain elongation, and increase of molecular weight M,(c) with concentration, should be observed. Deviation from the Flory theory is expected, since chain extension is in competition with ring formation, and because of the considerably higher molecular A2(C) = 442(0)(Mw(c) /Mw(o))-a (17) weights of the associates a decrease of the equilibrium constants, The third virial coefficient A3 is now expressed in terms of A2 as for instance by the osmotic pressure, appears conceivable. For follows this reason, the extent of reaction CY can no longer be expressed in terms of two equilibrium constants. The concentration deA3&C2 = g~(Apu,C)~ (18) pendence of CY for coupled telechelic polymers is shown in Figure where the factor gA is characteristic of the molecular architecture. 6. Knowing a,the expected molecular weight could be calculated Inserting (18) and (1 7) into (1 3) yields23 with the aid of eq 4 using fp = 2. The experimental molecular M, = l/Mapp[l + 2 A 2 ~ o ~ ~ M w ~ o ~ l ~ ~+M w ~ c ~ 1 1 - ~ c weights were derived from SLS data after correction for virial An ~ ~ * ~ ~ 2 ~ ~ ~ ~ ~ w(19) ~ ~ ~ coefficients I ‘ ~ ~as already w ~ ~ shown ~ 1 for 1 monofunctional - ~ ~ ~ 2 polymers. 1 exponent a = 0.2 now was used, which corresponds to long PS For linear polymers, theory24and experiment reveal g = 0.29 and chains in good solvents (Figure 2). The result is shown in Figure an exponent a = 0.2 in a good solvent if the molecular weight is 7, where again Mw(c) is normalized by Mw(0). Much smaller large enough. For lower molar masses (Le., M, < 20 X lo3) a values were found by SLS than predicted from the extent of was found to be a = 0.5 (see Figure 2).2s The real molecular reaction CY and the theory. The deviations are considerably larger weights Mw(c)can now be calculated from the scattered intensity, for the shortest primary chain. The reason for these deviations the concentration c, and the second virial coefficient A2(0)using is easily discerned. At low concentrations, Le., in the very dilute eq 19, applying an iterative method. The method is limited to region, the two functional groups of two different polymers are concentrations below ca. 5%. Because of experimental errors, on the average farther apart than the two ends of the same mainly in A2(0),which are considerable for low molecular weights, polymer. These theiefore react preferentially to form a ring. the method becomes inaccurate at higher concentrations. The Naturally, the ring closure decreases as the chain become longer.26 results for the two polymers M 0 1 and M 0 2 are shown in Figure 5 , where the M,(c) is normalized to Mw(0),which is the weight 5. Determination of the Ring Fraction average molecular weight of the 1:l mixture at zero concentration. Three procedures with increasing reliability are discussed. The polymer concentration is expressed in moles per liter which 5.1. Mean Field Approximation. The extent of ring formation allows comparison of the two polymers of different molecular can be estimated intuitively by a simple procedure. This consists weights. The result is in good agreement with the curve calculated which will give in evaluating an equivalent extent of reaction a,~, with Stockmayer’s formula (no ring formation) the measured (21) Yamakawa, H. Modern Theory of Polymer Solutions; Harper and molecular weight. This equivalent a,ffis, of course, smaller than Row: New York, 1971. the spectroscopically determined a. We may set (22) De Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell A2(O) = KAMW(O)-O
(16) where A2(0) is the real virial coefficient a t c = 0; this is approximately A2 for the pure polymer in dioxane without coupling reagent. The presence of additional low molecular weight dinitrophenol does not change A 2 ( 0 )significantly (see Table 11). Thus we write23
University Press: Ithaca, NY, 1979. ( 2 3 ) Burchard, W. Makromol. Chem. Makromol. Symp. 1990.39, 179. (24) Freed, K. F. Renormalization Group Theory of Macromolecules; Wiley: New York, 1987. ( 2 5 ) Huber, K.; Bantle, S.; Lutz, P.; Burchard, W. Macromolecules 1985, 18, 1461.
%ff
= CY (1 - a,)
(20)
(26) Semlyen, J. A. In Cyclic Polymers; Semlyen, J. A., Ed.; Elsevier Applied Science Publishers: London, 1986; pp 1-41.
3920
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 i.co I
Merkle and Burchard 5.3. Exchded Volume Effect. In the JS treatment the excluded volume was taken into account by means of the expansion factor as. Obviously this is only a rough approximation based on Flory’s uniform model of coil expan~ion.~ Excluded volume, however, disturbs Gaussian statistics,21such that the coil expansion at the chain ends is stronger than in the inner part of the coil. The end to end distribution function now has a minimum if R 0 (correlation hole) and this should decrease the probability of ring closure. Experimental evidence for this effect was found by who studied cyclization dynamics of labeled polystyrenes by means of fluorescence spectroscopy. The rate constant for cyclization was found to be smaller in good solvents toluene, benzene, or THF than in the 6’-solvent cyclohexane. Recently, Rubio and Freire30 performed Monte Carlo simulations to calculate the molecular weight dependence of cyclization probabilities for PDMS chains. Compared to the unperturbed chains, introduction of excluded volume decreased the cyclization probabilities. To examine the effect of excluded volume in the present experiments, which were carried out in the good solvent dioxane, we calculated the weight fraction of rings w, according to the JS treatment but now with the use of a distribution function, which reflects the excluded volume more accurately. That is the generalized DomMillis-Wilmers function, deduced from computer s i m u l a t i o n ~ ~and l - ~ derived ~ from RG theory by des C l o i ~ e a u x ~ ~ for long flexible chains
-
000 I
’
’
’
10-1
, , , , n
’
’
100
’
” ” ”
’
10‘
0
102
c [ d l Figure 8. Ring fraction as a function of polymer concentration for the BIl, (A)B12, and (V)BI3, calculated from the three telechelic PS: (0) mean field approximation; (m) BIl, (A) B12, and (V) BI3, calculated
from Jacobson-Stockmayer treatment. where a,is the fraction of reacted functional groups, which is involved in the ring formation, or in Flory’s terminology, which is w a ~ t e dLe., , ~ ineffective. This attempt to estimate a ring formation is clearly a crude approximation. That is, a,denotes just an average probability and is the same for all functional groups, no matter whether they become incorporated into a chain or a ring. 5.2 The Jawbson-StOckmeyer Treatment. Assuming Gaussian statistics for the distribution of end to end distances, Jacobson and Stockmayer (JS)27developed a more reliable procedure to determine the probability of ring closure in linear macromolecules. When the polymer system contains equivalent quantities of the two monomers, the ring fraction by weight w,, is given by the following expression: w,
= 2(B’/4dx2/3/2)
(21)
with
B’ = (1 /N0)(3/27rv)312( 1/ b 3 )
(22)
W(R) = CRs exp[-(R/u)‘] where C is a normalization factor and is given by
c l = (#+l/t)r((s + i)/t)
+
u2 = ( R 2 ) r ( ( s l)/t)/I’((s
in which M’is the number of both monomers in rings, M the total number of monomers in the system, and [A],,, the concentration of reacted functional groups in the ring fraction. Following JS, x is connected with a and w, a=x
+ (1 - x)w,
(25)
Effective bond lengths b = 0.365, b = 0.414, and b = 0.487 nm for the three polymers BI1, B12, and B13 were used for reasons which will be discussed below. A variation of b was expected since excluded volume has not yet been taken into account. However, the effective bond length should then be larger than the effective bond length in a 6’-solvent, be = 0.487 nm. The ring fraction w, was obtained by varying x in eq 21 until an a in eq 25 was obtained that coincided with the measured one. The results are shown in Figure 8 together with values obtained from the mean field approximation, after being converted into w, = a a,. (27) (a) Jacobson, H.; Stockmayer, W. H. J . Chem. Phys. 1950,18, 1600. Stockmayer, W. H. J. Chem. Phys. 1950, (b) Jacobson, H.; Beckmann, C. 0.; 18, 1607. (28) Truesdell, C. A. Ann. Math. 1945, 46, 144.
(27)
where r denotes the gamma function and s, t, and u are adjustable parameters. The parameter u is related to the mean square end-to-end distance (R 2 ) by
and the Bose-Einstein function
Here No is Avogadro’s number, v the number of backbone C atoms in the smallest possible ring, b the effective bond length, c the total concentration of both “monomers” in moles per unit volume, and x the fraction of reacted end groups in the chain fraction, and the summation index n is the number of units of each kind in the associate. The summation index n in the Bose-Einstein function runs from one to infinity and a table for various x was given by T r u e ~ d e l l . ~ ’The ~ ~ ~ring fraction is defined as
(26)
+ 3)/t)
(28)
For further evaluation we used s = 2.29 and t = 2.43, derived from computer simulations and renormalization group t h ~ r y . ~ ~ , ~ ’ To determine the probability of ring closure for the given polymer system, we followed the JS2’ scheme, but now with the distribution of eq 26 instead of a Gaussian distribution. We used now ( R2) = (~n)’.’’~b,~, with 6, the effective bond length in good solvent and the exponent 1.176 = 2 X 0.588, where 0.588 is the common exponent in the Rg-Mwrelationship for infinitely long polymers in good solvents. The above expression for the end to end distance ( R 2 )implies the same power law behavior of the expansion factors a R 2 and as2 in z (a2 z2I5),where z is the excluded volume parameter, that was found for z > 2 by computer simulation^.^^ From comparison with experimental data of radii of gyration Rgof linear polystyrene in toluene2’ b, was found to be b, = 0.34 nm in open chains. Straightforward calculation leads for rings to the following result
-
w,
= 2(B’/c)(p(x2,1.93)
(29)
with The difference to the JS treatment consists in the dependence on molecular weight, occurring in the exponent for Y, the exponent (29) (a) Winnik, M. A. In Cyclic Polymers; Semlyen, J. A., Ed.;Elsevier Applied Science Publishers: London, 1986; pp 285-348. (b) Winnik, M. A. Acc. Chem. Res. 1985, 18, 73. (c) Winnik, M. A.; Sinclair, A. M.; Beinert, G. Con. J . Chem. 1985, 63, 1300. (30) Rubio, A. M.; Freire, J. J.; Horta, A.; PiErola, I. F. Macromolecules 1991, 24, 5167. (31) Fujita, H. Polymer Solurions; Elsevier: Amsterdam, 1990. (32) Mazur, J. J . Res. Natl. Bur. Stand. 1965,69A, 335; J. Chem. Phys. 1965, 43,4354. (33) Domb, C.; Gillis, J.; Wilmers, G. Proc. Phys. SOC.(London)1965, 85, 625. (34) Des Cloiseaux, J. J . Phys. (Paris) 1980, 41, 223. (35) Le Guillou, J. C.; Zinn-Justin, J. Phys. Reu. Lett. 1977, 39, 95.
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 3921
Amino-Terminated Polystyrenes
TABLE III: Concentrations of Polymer, Extent of Reaction (a)from UV Spectroscopy, Weight FractioG of Rings w, Obtained from JS Treatment and from Excluded Volume, and Molecular Weights (x10-3)a
(a) Polymer BI1 MP = 17500 M.. ~~
c.n/L 0.900 1.360 1.810 2.260 4.520 9.040 13.560 18.070 22.590
a
JS
EV
0.780 0.820 0.830 0.835 0.855 0.870 0.890 0.890 0.910
0.748 0.765 0.772 0.802 0.763 0.694 0.650 0.586 0.582
0.752 0.769 0.775 0.756 0.762 0.688 0.664 0.594 0.566
LS 18.00 18.30 18.70 18.90 22.50 28.10 33.00 36.10 43.00
JS
EV
18.49 18.84 19.18 19.50 21.45 26.34 32.54 39.63 49.61
18.47 18.80 19.13 19.49 21.33 26.21 32.57 40.14 49.90
(b) Polymer BIZ, MP = 35000 (4
c,g/L 0.317 0.634 1.270 2.540 3.170 6.340 12.680 19.020 25.360
M w
a
JS
EV
LS
JS
EV
0.500 0.560 0.620 0.660 0.710 0.740 0.760 0.800 0.810
0.436 0.469 0.517 0.461 0.509 0.438 0.325 0.290 0.243
0.487 0.501 0.496 0.468 0.511 0.434 0.318 0.273 0.227
36.40 38.00 40.00 44.20 51.20 56.40 71.70 97.80 120.00
36.37 37.30 39.25 42.82 45.23 55.60 74.07 97.62 113.80
36.47 37.42 39.29 43.12 45.60 56.52 76.15 98.50 114.75
(c) Polymer BI3, MP = 50000 0,
c,g/L 0.430 0.540 1.070 2.140 3.210 4.290 5.360 10.710 21.340 32.140
a
0.335 0.365 0.418 0.500 0.540 0.562 0.590 0.658 0.723 0.740
JS 0.226 0.229 0.225 0.240 0.221 0.202 0.194 0.147 0.108 0.077
M
EV 0.229 0.235 0.239 0.244 0.241 0.220 0.212 0.167 0.117 0.087
LS 50.20 58.70 67.90 68.30 73.30 74.70 80.50 97.00 120.10 160.60
W
JS
EV
51.58 52.19 54.97 61.59 67.17 72.05 77.68 99.09 137.15 150.47
51.81 52.37 54.97 60.61 66.43 71.06 76.39 98.96 132.82 150.91
a JS = Jacobson-Stockmayer treatment; EV = excluded volume; LS = light scattering.
in the -Einstein function and in the prefactor B. Both changes decrease the ring fraction compared to the JS treatment. The evaluation of the ring fraction from spectroscopical data was made as shown for the Gaussian rings. To get a good description of the experimental data we now had to use effective bond lengths b, = 0.1 56 nm for the polymers BI1 and B12, and b, = 0.170 nm for B13 in the treatment as outlined in section 6. The results are listed in Table 111.
6. Determination of Molecular Weights from w, The JS treatment provides the possibility to calculate molecular weight distributions in the ring and the chain fraction only from spectroscopical data. Here we confine ourselves to the evaluation of weight average molecular weights M,(c), since a comparison of M,(c) obtained from LS with those from the treatment below enables us to test the validity of the calculated ring fraction w,. 6.1. JS Treatment. The weight average molecular weight of the associating system consisting of rings and chains is given by Mw(c) = W r M w r + (1 - wr)Mwc (31) where M,, is the weight average molecular weight of the ring fraction and MWc is that of the chain fraction. M,, is given by
M,, = ( cR,n2/ C R , n ) M ~= n= I
((p(X2,1,2)/(p(x2,3/2))M~o
(32) Here R, is the number of n-meric rings2'
(33)
where Vis the volume of the system, n denotes the number of repeating units (consisting of one polymer and one coupler molecule) with the molecular weight Mr0= M p+ Mk.The molecular weight of the chain fraction M,, is given by (4), but now we have to use x, the extent of reaction in the chain fraction, instead of CY, the overall extent of reaction. This molecular weights were then compared with those obtained from evaluation of LS data. The most sensitive and uncertain parameter in this calculation is the effective bond length b, slight changes in b cause an immense change in w, and therefore also in M,(c). Thus we used b as an adjustable parameter and best agreement of Mw(c)with M,(c) from LS was found with b = 0.365, 0.415, and 0.487 nm for polymers BI1, B12, and BI3, respectively. These data are systematically larger than b, = 0.34 nm found for the open chain with excluded volume. The increase with molar mass was expected since here the EV has not yet been taken into account. 6.2. Excluded Volume Effect. We can use the same equations as in section 6.1; all we have to change are the exponents in the Bose-Einstein functions and the prefactor B. Once again the effective bond length b was chosen to be the adjustable parameter. Now best agreement with molecular weights Mw(c)from LS was found with b, = 0.156 nm for the telechelics BI1 and B12 while be, = 0.170 nm had to be used for B13, which are much lower than be, = 0.34 nm obtained from the open chains.
7. Discussion 7.1. Coupling of Monofunctional Polymers. The treatment of monofunctional polymers has shown that it is possible to predict the molecular weight of the associates at any given polymer concentration and temperature. All we have to know is the molecular weights of the primary PS chain, which is well defined when the polymer is prepared by anionic polymerization, and the equilibrium constants which can be derived from spectroscopy. The agreement between experiment and theory also shows that an association due to hydrogen bonding between hydroxyl and carboxy groups of the coupler does not play an essential role as was suggested by Young et 7.2. Coupling of Tdechelic Polymers. The coupling of telechelic polymers showed that besides chain elongation, ring closure is appreciable. The deviation between mean field approximation and the JS treatment is most pronounced for the low molecular weight sample. This fact reflects the strong molecular weight dependence of ring formation which is neglected by the mean field approximation. The most reliable values of w, according to the original JS treatment (no excluded volume) were obtained with effective bond lengths b = 0.365,0.415, and 0.487 nm for the polymers BI1, B12, and B13, respectively. The increase of b with molecular weight reflects the increasing chain expansion in the good solvent dioxane. However, the effective bond lengths found here are smaller than that obtained from the characteristic ratio c" 10 for linear polystyrene?' be = 0.487 nm in cyclohexane. Thii can mean lower unperturbed dimensions of linear PS in dioxane which gave be, = 0.34 nm for the open chains. All the chains used here are larger than M , = 104 and chain stiffness should have no longer influence on the unperturbed effective bond length. There was a great deal of work on the effect of chain stiffness on ring formation in literature. Treatments on wormlike and simulations for chains composed of cylindersa and for PDMS chains based on the RIS model with and without consideration of long-range interactions30 have shown a maximum for the ring closure probability when the chains are very small (about 20
-
(36) Young, R.N.;Quirk, R. P.; Fetters, L. J. Adu. Polym. Sci. 1984, 56,
28.
m
n= 1
R, = BVx2"n-s/2
(37)Brandrup, J.; Immergut, E. H. Polymer Hundbook, 2nd 4.Wiley: ; New York, 1975. (38) Yamakawa, H.; Stockmayer, W. H. J . Chem. Phys. 1972,57, 2843. (39)Shimada, J.; Yamakawa, H. Mucromolecules 1984, 17, 689. (40) Post, C.B. Biopolymers 1984, 23, 601.
J. Phys. Chem. 1992,96. 3922-3926
3922
skeletal bonds for PDMS or polymethylene chains). Although PS is much stiffer than P M or PDMS the chain lengths of the present polymers are too long for showing this effect. The ring fraction is further decreased if the excluded volume is taken into account, because excluded volume disturbs Gaussian statistics. To reach the Mw(c)measured by LS it was necessary to counterbalance that by a further reduction of the effective bond length. The measured Mw(c)were obtained in the fit with be, = 0.156 nm for PS BI1 and B12 and b, = 0.17 nm for sample B13, which is smaller than be, = 0.34 nm in toluene. In contrast to the original JS treatment, the effective bond length is now no more that strongly molecular weight dependent, since the effect of excluded volume is taken into account. Slight deviations are still present, since throughout relations were used which are only valid in the asymptotic region of very high molecular weights. The molecular weights of the samples investigated here are below that limit, but they are too high for showing striking features of chain stiffness. The value of the effective bond length be, found from the modified JS treatment be, = 0.16 f 0.01 nm is considerably smaller than the corresponding value for the open chains with b, = 0.34 nm. At present we are not able to give a conclusive explanation. One possibility may be that the effect of excluded volume is overestimated for rings if the same asymptotic behavior?' as= 1 . 5 3 ~ * /with ~ , the same prefactor is taken for the rings and open chains. Comparison of our results with those obtained by synthesis of covalent polystyrene ring polymer^^^^^^*-^^ is quite difficult. Although the two systems seem to be very similar there are some (41) Geiser, D.; Hijcker, H. Macromolecules 1980, 13, 653; Polym. Bull. 1980, 2, 591. (42) Vollmert, B.; Huang, J. X . Makromol. Chem. Rapid Commun. 1980, I , 332; 1981, 2, 467. (43) Roovers, J.; Toporowski, P. M. Macromolecules 1983, 16, 843.
essential differences which complicate a direct comparison. First, in the present system the extent of reaction depends on the concentration of functional groups, while in the synthesis of polystyrene rings a = 1 may be assumed. Second, according to the method the covalent ring polymers are prepared, the ratio r of the functional groups is not r = 1, as in our case, but is increasing with the cyclization since the coupling reagent is added slowly to a carbanionic PS solution. Third, the most interesting point is the difference in reaction mechanism. Treatment of ring closure reaction for irreversible systems is much more complicated and to our knowledge this problem has been solved only approxi-
Acknowledgment. The idea for this paper occurred many years ago to Professor W. H. Stockmayer. We took up this idea with great pleasure but realized during this work how difficult the problem is. We are most grateful for the suggestion of this work and the many very useful discussions on the problem of ring formation. We also thank with warmth Professor J. J. Freire who informed us of his Monte Carlo calculations prior to publication, for his valuable personal remarks, and Professor K. F. Freed for fruitful exchange of ideas. This work was not conceivable without the experience of and the help by Dr. P. Lutz in Strasbourg. The project was financially supported by the SFB 60 of the Deutsche Forschungsgemeinschaft. Registry No. I, 139896-75-6. (44) Stepto, R. F. T. In Developments in Polymerization 3; Haward, R. N., Ed.; Applied Science: Barking, England, 1981; pp 81-141. (45) Gordon, M.; Temple, W. B. Makromol. Chem. 1972, 160, 263. (46) Gordon, M.; Temple, W. B. Makromol. Chem. 1972, 152, 277. (47) Stepto, R. F.T.; Waywell, D. R. Makromol. Chem. 1972, 152, 263. (48) Kilb, R. W. J. Chem. Phys. 1958, 62, 969. (49) Cowie, J. M. G.; Worsfold, D. J.; Bywater, S. Trans. Faraday Soc. 1956, 52, 165 1.
(50) Berry, G. C. J . Chem. Phys. 1966, 44, 4550.
Density Profile of Terminally Attached Polymer Chains Paul Venema Department of Physical and Macromolecular Chemistry, Gorlaeus Laboratories, University of Leiden, P.O. Box 9502, 2300 RA Leiden, The Netherlands
and Theo Odijk* Department of Polymer Technology, Faculty of Chemical Engineering, Devt University of Technology, P.O. Box 5045, 2600 GA Deut, The Netherlands (Received: August 28, 1991)
Scaling theory hypothesizes a stepfunction profile for the segment density of polymer chains terminally attached to a planar wall. Using a self-consistent-field theory we give a perturbation analysis of the first-order correction to the step-function profile in order to gauge the impact of a possible tail. The new profile decays smoothly to zero without a discontinuity in the derivative (except near the wall). The segment density profile as scaled by the amplitude of the step-function profile ~ , B a dimensionless parameter and z* the distance from the wall scaled by has a tail that decays essentially as ~ / B Z * with the step length of the step-function profile. A scaling analysis would yield z * - " / ~ .
Introduction We consider long and monodisperse, flexible polymer chains in a good solvent, which are chemically attached by one end to a (planar) wall. We assume that the density of grafts at the surface is high enough, so that the chains tend to stretch away from the surface. Our concern is to investigate the density profile 4 2 ) of the segments as a function of the distance z from the wall. This problem has been addressed by several authors. Scaling arguments of Alexander' and de Gennes* hypothesize a step(1)
Alexander,
s. J. Phys. (Paris) 1977, 38, 977. 0022-3654/92/2096-3922S03.00/0
function profile for the segment density. The theory of Milner, Witten, and who use the analogy of a strongly stretched chain to the trajectory of a classical particle, predicts a parabolic profile. An identical result was also found independently by Zhulina et al? by a " i z a t i o n of the conformational free energy (2) De Gennes, P. G. J . Phys. (Paris) 1976, 37, 1443; Macromolecules 1980, 13, 1069. (3) Milner, S . T.; Witten, T. A.; Cates, M. E. Macromolecules 1988.21, 2610. (4) Zhulina, E. B.;Borisov, 0. V.; Pryamitsin, V. A. J . Colloid Interface Sci. 1990, 137, 495.
0 1992 American Chemical Society
