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Experimental Observation of Crossover from Noncondensed to Counterion Condensed Regimes during Free Radical Polyelectrolyte Copolymerization under High-Composition Drift Conditions Tomasz Kreft and Wayne F. Reed* Physics Department, Tulane UniVersity, New Orleans, Louisiana, 70115 ReceiVed: February 23, 2009; ReVised Manuscript ReceiVed: April 30, 2009
By combining detailed online kinetics of comonomer consumption with light scattering, viscosity, and conductivity data, experimental detection of changing degrees of counterion condensation was achieved by using comonomers of widely separated reactivity ratios that produced large composition drifts during synthesis. Endproducts of such syntheses contained mixtures of chains of widely varying linear charge density. Evidence of a smooth transition from noncondensed to counterion condensed regimes was found during individual synthesis reactions of copolymeric polyelectrolytes, or “copolyelectrolytes”, from the changing slope of conductivity versus [ionic comonomer] incorporated in the high-composition drift syntheses. From this, a model was used to connect the latter slope behavior to fractional ionization of counterions. With this modeled fractional ionization versus the instantaneous fractional amount of the charged comonomer incorporated into copolyelectrolyte, which is directly and continuously monitored during synthesis, it was possible to compute linear charge density ξ and the corresponding average ξ distribution. Introduction It was recently demonstrated that monitoring molecular weight, comonomer composition drift, and solution conductivity σ during the synthesis of copolymeric polyelectrolytes (henceforth “copolyelectrolytes”) helps connect the previously disparate fields of synthetic chemistry of copolymers and physical chemical properties of polyelectrolytes.1 The relationship between diminishing σ and charged comonomer incorporation was monitored and provided novel data on counterion condensation (CC), which occurs gradually over a broad composition regime. The specific conductivities of the copolyelectrolytes were also found to be molecular weight independent, in agreement with most theories2 and previous experiments.3 The field of polyelectrolyte solution properties4 has been intensively studied theoretically and experimentally. Long-range electrostatic forces make the polymer excluded volume problem5 very difficult, and have led to the concepts of “electrostatic persistence length and excluded volume”.6 The electrostatic force also gives rise to such effects as angular scattering maxima of neutrons,7-9 X-rays,10 and light,11-14 including under shear15 suggestive of liquid-like correlations, and also causes the “electroviscous effect”,16,17 wherein reduced polyelectrolyte viscosity increases under isoionic dilution in very low ionic strength solutions. Analysis of the copolyelectrolytes in this work was done at high enough ionic strength (0.1 M) to suppress these types of effects, so as to concentrate on the copolymer characteristics. CC in polyelectrolyte solutions is another phenomenon18,19 that has also been the topic of extensive analytical and numerical investigations,20,21 but for which far fewer detailed experimental works exist.22,23 CC occurs when the electrostatic potential energy between charge sites on a polymer chain and counterions at the intermonomer backbone distance exceed kT. ξ is defined * To whom correspondence should be addressed. E-mail: wreed@ tulane.edu.
in this context as the number of elementary charges per Bjerrum length lB. The previous work furnished high resolution data intrinsically linking comonomer composition to copolyelectrolyte ξ, and provided grist for theoretical descriptions.1 A first, elementary model for obtaining ξ from these data was also proposed. In that work1 the comonomers were acrylamide, Am, (electrically neutral) and [2-(acryloyloxy)ethyl]-trimethylammonium chloride (Q9), with a single cationic charge. The reactivity ratios of Q9 and Am were close enough to each other (rQ9 ) 0.47 and rAm ) 1.10), that there was relatively little composition drift in any of the synthesis reactions, which covered a wide range of [Q9]0/[Am]0 starting values. It was found in all reactions that σ versus [Q9] was linear; that is, the loss of σ per Q9 monomer incorporated into polyelectrolyte was independent of conversion and small amounts of composition drift. At low starting [Q9]0/[Am]0, that is, low charge density in the copolyelectrolyte chains produced, the decrease in σ during the reaction was solely due to monomer incorporation into copolyelectrolyte chains. At higher [Q9]0/[Am]0 the decrease in σ during the reaction was higher due to counterion condensation. None of the composition drifts, however, was large enough to cross through the broad noncondensed to condensed counterion transition regime, but the slopes of the linear σ vs [Q9] data increased monotonically as [Q9]0/[Am]0 increased in the experiments. The object of this work is to monitor the changing degree of counterion condensation within individual synthetic reactions, when there is a strong composition drift. To achieve this, two monomers with quite different reactivity ratios were chosen, the salt form of 4-vinylbenzenesulfonic acid sodium salt, VB, and Am; reactivity ratios were determined from ACOMP data to be rVB ) 2.14 and rAm ) 0.18. With the crossover regime mapped, it is possible to use a model to determine the average ξ distribution of the copolyelectrolyte endproduct, in addition to the average composition, molecular weight, and intrinsic
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TABLE 1: Summary of Copolymerization Experiments expt no.
VB% (moles)a
reactor CVB,0 (g/mL)
detector CVB,0 (g/mL)
Mw (g/mol) (f ) 0.8)
ηr (cm3/g) (f ) 0.8)
1st order VB rate (1/s)
1st order Am rate (1/s)
1 2 3 4 5 6 7 8 9 10 11 12
100 80 70 60 50 40 30 25 20 15 10 5
0.0103 0.0576 0.0517 0.0435 0.0370 0.0297 0.0221 0.0185 0.0149 0.0110 0.0074 0.0037
0.00041 0.00046 0.00036 0.00043 0.00037 0.00030 0.00022 0.00022 0.00022 0.00013 0.00009 0.00005
N/A 562 000 452 000 344 000 315 000 254 000 187 000 160 000 160 000 160 000 N/A 250 000
N/A 157 143 115 105 100 97 81 60 65 75 81
0.00076 0.00066 0.0006 0.00044 0.00036 0.00039 0.00037 0.0004 0.00044 0.00047 0.00055 0.00073
N/A 0.00073 0.00051 0.00043 0.00027 0.00028 0.00022 0.0002 0.0002 0.00016 9.20 × 10-05 3.70 × 10-06
a
[VB]0 + [Am]0 ) 0.36 M/L for all experiments except no. 1, where [VB]o ) 0.05 M/L.
viscosity distributions. In these high-drift experiments the ξ distribution is no longer simply proportional to the composition distribution, as was found in the low drift cases of ref 1. Materials and Methods A series of experiments at different starting [VB]0/[Am]0 ratios were performed. These are summarized in Table 1. VB was copolymerized with Am and the reaction was initiated with 2,2′-azobis(2-amidinopropane)dihydrochloride (V50). All copolymerization reactions were conducted in 0.005 M NaCl solution at 60 °C. Water was deionized and filtered with 0.22 µm filter in a Modulab UF/UV system. The principle of ACOMP and system specifics have been reported previously in detail.24,25 The custom built ACOMP system in this work used a five pump, two stage dilution in a low and high pressure mixing chambers (LPMC and HPMC), yielding from 77× to 125× dilution, depending on the amount of monomer concentration. This approach was recently described in ref 26. The ACOMP dilution solvent was aqueous 0.1 M NaCl, which was a high enough ionic strength to suppress most of the electrostatic interactions between copolyelectrolyte chains, and to determine Mw and [η]w. Detectors comprised a Brookhaven Instruments Corporation (BI-MwA) multiangle light scattering photometer (MALS), a Shimadzu RID-10A differential refractometer (RI), a custombuilt single capillary viscometer,27 and a Shimadzu photodiode array SPM-20A UV/visible spectrophotometer (UV). The in situ conductivity probe (Jenway) directly measured conductivity σ(t) of the reactor liquid, which was kept at 0.005 M NaCl. As seen in Table 1, the total amount of VB and Am was always less than 5% by weight in all reactions. This is similar to experiments in ref 1, where synthesis was carried out in a dilute enough regime (∼3% weight fraction of total comonomer in aqueous solution with 5 and 25 mM NaCl), that the growing copolyelectrolyte chains did not alter electrophoretic mobilities of the monomeric and polymeric species and their counterions. In this work [x] denotes concentration of reagent x in mol/L, whereas cx represents concentration in g/cm3. A useful conversion from starting molar concentration ratio [VB]0/[Am]0 to starting molar mass concentration cVB,0/(cvb,0 + cAm,0) is
cVB,0 ) (cVB,0 + cAm,0)
1 mAm[Am]0 1+ mVB[VB]0
(1)
where the monomer formula masses are mVB ) 206.19 g/mol and mAm ) 71.08 g/mol. In some instances in this work it is
Figure 1. Different types of comonomer conversion for experiment 9: ftotal,molar, ftotal,mass, fVB, fAm.
preferable to use molar concentrations and in other contexts (e.g., computation of molecular weights and intrinsic viscosity) the mass concentration is preferable. Results and Discussion Figure 1 shows fractional comonomer conversions for experiment 9; total monomer conversion expressed in both mass and molar terms, ftotal,mass and ftotal,molar, respectively, as well as the individual fractional monomer conversions of VB and Am (for the individual components the fractional molar and mass conversions are equal). For reference, a convenient relationship between ftotal,mass and ftotal,molar is given using total initial molar and mass concentration of comonomers [M]0 and c0, respectively.
dftotal,mass [M]0 ) (Finst,VB(mVB - mAm) + mAm) dftotal,molar c0
(2)
A striking two phase conversion is seen for Am, wherein it copolymerizes with VB in a first phase, then after VB’s first order exhaustion, continues to homopolymerize, thus producing a blend of highly compositionally disperse copolyelectrolyte and homopolymeric pAM. This feature of the VB and Am copolymerization was reported earlier.28 Table 1 also gives the first order monomer decay rates for VB and Am for each reaction (for Am the rate is for the first phase in the case of two phase reactions).
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Figure 3. Instantaneous fractional molar composition of VB, Finst,VB vs ftotal,molar.
could have interesting consequences in producing terpolymeric polyampholeytes composed of Am, Q9, and VB. Namely, the composition could be controlled to tailor Mw to desired regimes. Coupled with semibatch addition of reagents even more control over Mw and composition could be achieved. Figure 2b shows reduced viscosity ηr,w vs ftotal,molar for two of the Table 1 experiments. The values are closer together and the signal noisier than the corresponding Mw data, so only two data sets are shown. The main feature is that ηr,w vs ftotal,molar follows the Mw trend, and, since ηr,w is directly related to molecular weight, it serves as an independent cross-check of the tendency of the latter to decrease as conversion increases. Figure 3 shows the composition drift for several of the experiments in Table 1. The drift is expressed in terms of the average instantaneous molar fraction of VB in chains formed at any instant of conversion Finst,VB, defined as Figure 2. (a) Cumulative weight average molecular weight Mw vs total molar conversion ftotal,molar for several experiments from Table 1. Figure 2b. Cumulative weight average intrinsic viscosity [η]w vs total molar conversion ftotal,molar for several experiments from Table 1.
Figure 2a shows weight average molecular weight Mw vs total molar monomer conversion ftotal,molar during synthesis for several reactions in Table 1. Except for early conversion, Mw decreases linearly with conversion, as is typical for free radical polymerization when there are no significant chain transfer reactions and the initiator is long-lived. The half-life of V50 is 10 h at 56 °C in water (http://www.wako-chem.co.jp/specialty/waterazo/ V-50.htm). A comprehensive method was recently presented29 for obtaining true copolymer Mw during ACOMP monitoring of copolymer reactions from the apparent molecular weight Map, obtained by direct extrapolation in the standard Zimm method.30 In this work the dn/dc of the comonomers are large and positive, and close enough together (0.192 cm3/g for pAM and 0.175 cm3/g for pVB) that even under high-composition drift conditions Mw is very close to Map. As seen in Figure 2a and Table 1, Mw increases as the starting fraction of VB increases. This is the opposite of what occurred in the Q9/Am case, where increasing the starting (cationic) Q9 fraction led to much smaller Mw. This would suggest that the ratio of propagation to termination coefficients for the individual monomers increases as (kp/kt)VB > (kp/kt)Am > (kp/kt)Q9 This trend
Finst,VB )
d[m]VB d([m]VB + [m]Am)
(3)
A recently established method for obtaining the concentration of each comonomer via analysis of full UV spectra every second during synthesis was used.31 All experiments at low starting [VB]0/[Am]0 (up to 0.5) exhibited high drift, which was limited by the starting value. At all starting values of [VB]0/[Am]0 e 0.2, the VB was actually totally consumed before total comonomer conversion was complete, as in Figure 1, leading to the mentioned blends of copolyelectrolyte of broad composition distribution, and neutral pAm, produced after the VB was exhausted. For reactions with higher starting VB there is no detectable two phase process. Within these individual high-drift experiments counterion condensation transitions were found that resemble the ones found among multiple low-drift reactions in the Q9-Am system (In the Q9 system there was no detectable condensation within individual reactions). Figure 4 is the primary evidence for the crossover from the noncondensed to the condensed counterion regime. It shows σ vs [VB] for several of the high-drift experiments, which is nonlinear in all cases shown. This is due to the composition drift during the reaction (see corresponding drifts in Figure 3) crossing from the counterion condensed to noncondensed regimes. The inset to Figure 4 shows that σ vs
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Figure 4. Total solution conductivity σ vs [VB] for several experiments in Table 1. The nonlinear nature is direct evidence of different degrees of counterion condensation as the composition drifts. The inset shows σ vs [VB] for a very low drift experiment (3) vs [VB] and σ vs [Q9] for low drift experiment from ref 1. The linear behavior of these inset curves shows there is no varying degree of counterion condensation during the reaction. The conductivity on both y-scales in the inset is mS/cm.
[VB] is linear when composition drift is low and stays within one of the regimes; for example, experiment 3 which stayed within the high Finst composition regime, for which all chains have condensed counterions. The other low drift experiments shown in Figure 3 likewise showed linear σ vs [VB]. Also shown in the inset is another example of linear σ vs [Q9] taken from previous experiments with the low drift comonomer combination Q9/Am.1 It was previously shown in ref 1 that all starting values of [Q9]0/[Am]0 led to linear σ vs [Q9], due to low composition drift. This linearity also demonstrated the independence of copolyelectrolyte chain electrophoretic mobility from Mw and also showed that the changing viscosity of the (low polymer concentration) solution in the reactor did not measurably affect electrophoretic mobility. The model-independent data of Figure 4 require an increasing amount of counterion condensation with increasing fraction of VB in the chain. If there were no counterion condensation and the specific conductivity of the copolyelectrolyte chains Σcp were constant and independent of Finst,vb (i.e., independent of linear charge density) then the slopes of σ vs [VB], dσ/d[VB], would be constant vs [VB] and the same among experiments. If there were no condensation and Σcp increased with increasing Finst,vb, as would be expected, then dσ/d[VB] would increase vs conversion in those reactions where Finst,VB decreases vs conversion. Finst,vb decreases vs conversion for all the higher number experiments in Figure 3 but dσ/d[VB] decreases vs conversion for these experiments, leading to the conclusion that CC increases more rapidly than Σcp with increasing Finst,vb. Note that the reactions occur from right (high [VB]) to left (low [VB]) in Figure 4. A visualization of the crossover regime is given in Figures 5a and 5b. Figure 5a uses slopes of σ vs [VB], dσ/d[VB], taken over discrete segments of the individual reactions shown. It is seen that the slopes increase as Finst,VB increases, which is due to the counterion condensation as linear charge density ξ increases. At low amounts of VB the slope is low since the change in σ is due to putting charged VB monomers into copolyelectrolyte chains, where they have less electrophoretic mobility than free VB, but their counterions remain free to move in an applied field. For the higher amounts of VB, counterions condense onto the polymer as VB is incorporated into the chain,
Figure 5. (a) Slope of σ vs [VB] for several experiments from Table 1. The increasing slope vs [VB] for each individual experiment shows an increasing counterion condensation at high fractional composition of VB. Figure 5b. Slope of σ vs [VB] for several experiments from Table 1 for average from experiments in Table 1 and for an individual experiment (no. 5), and, for the right-hand y-axis, the slopes of σ vs [Q9] from ref 1.
taking a much larger amount of solution conductivity away. As found previously the transition from noncondensed to counterion condensed is gradual and does not involve any sharp transition as is predicted by the simplest theory. A peculiar feature of the data in Figure 5a is that there is some separation in the trends of the conductivity slopes between the low starting [VB]0/[Am]0 and the higher values. The lower [VB]0/[Am]0 data cluster together at higher slope values than the cluster of slope values for the higher starting [VB]0/[Am]0. There is no compelling argument for this difference in trends at this time, other than the error bars are high enough between the clusters that they could be considered to fall within each others’ bounds. Taking derivatives of experimental data with noise always leads to much larger uncertainties in the derivatives than in the primary data. Shown in Figure 5b are the average conductivity slope data for most experiments in Table 1 and, for contrast, the same data obtained by averaging over multiple individual experiments in ref , where σ vs [Q9] was linear for each experiment. All
Regime Crossover under High-Composition Drift conductivity measurements were at T ) 60 °C. The specific conductivity Σ of Q9 is ΣQ9 ) 77 mS-L/cm-mole and that of VB is ΣQ9 ) 57 mS-L/cm-mole. It is interesting that the intrasynthesis slopes for the Table 1 VB/Am copolymers follow a similar trend as those found for the multiple individual experiments using Q9/Am, although the values and shape are different. In these conditions the feature of a broad transition regime in composition is similar. Q9 and VB have the same contour length for their polymerizable vinyl groups that constitute the copolyelectrolyte backbone, although the cationic charge on Q9 is further removed from the backbone than the anionic charge for VB, and each charged monomer has a different counterion. Slope values for reactions 2 and 3 are absent from Figure 5b because the conductivity probe was near saturation. Features concerning σ common to all reactions were (i) solution viscosity increasing during the reaction did not decrease electrophoretic mobility of polymeric and monomeric species and (ii) molecular weight does not change mobilities. Average Composition and Linear Charge Density Distributions. Polymers can be characterized by a number of distributions associated with their properties. At the simplest level, polymers are characterized by their molecular weight distribution. Several additional distributions that can be used in characterization include intrinsic viscosity [η], chemical composition, and linear charge density ξ. In this work and the related, previous one, it was seen that ξ was independent of the molecular weight distribution. If there were no counterion condensation, then ξ would simply be proportional to Finst,VB, the instantaneous fractional molar composition of VB. In the previous work ξ was always proportional to Finst,Q9, for any giVen synthesis, because there was very little composition drift. In the data gathered here, however, the deviation between the ξ and Finst,VB distributions can be distinguished. Unlike most ACOMP derived quantities, however, a model must be used to relate the conductivity to ξ and Finst,VB. It is emphasized that there are many subtleties inherent in treating solution conductivities involving polyelectrolytes, and the following is a simple sketch for interpreting the marked nonlinearities of σ vs [VB]. The data presented here can serve as grist for more refined and rigorous models. One connection is made via the following model for interpreting the relationship between total, measured reactor solution conductivity σ(t), and ξ and Finst,VB. Following the procedure of ref 1 and substituting [VB] for [Q9];
σ(t) ) σNaCl + [VB](t)ΣVB + ([VB]0 [VB](t))(φ(t)ΣNa + Finst,VB(t) φ(t)ΣCP) (4) where σNaCl is the conductivity due to added salt in the reactor at the reaction temperature, which is constant during the reaction, φ(t) is the fraction of noncondensed Na+ counterions, ΣVB is the specific conductivity of free VB (it includes both the conductivities of the anionic monomer and its free Na+ counterions), ΣNa is the specific conductivity of Na+, and ΣCP is the specific conductivity of VB in the polyelectrolyte (i.e., the conductivity of a VB monomer incorporated into a copolymer chain). In ref 1 it was assumed, as a first approximation, that changes in σ(t) due to changes in ΣCP are much smaller than changes in σ(t) due to counterion condensation and to putting free VB into chains of an average ΣCP.
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Figure 6. Fraction of VB groups in polyelectrolyte chains that are ionized, φ, obtained from computations by eq 6 for experiments in Table 1, then averaging. Fit is to sigmoid of eq 7.
σNaCl ) 1.09 mS/cm was obtained from a linear fit to σ(0) vs [VB]0 points for each experiment in Table 1. It is noted that there was always a small spike in σ(t) upon adding V-50 that is subtracted out in eq 4. ΣNa ) 43.8 mS-L/cm-mole and ΣCP ) 41.5 mS-L/cm-mole were found by simultaneous application of the following assumed limiting condition.
φ ) 1, Finst,VB f 0 φ ) 0.36, Finst,VB ) 1 ΣVB ) 57.3 mS-L/cm-mole ( 4%, was found by averaging over all experiments the values found individually for each experiment from the initial data just before each reaction started, from
ΣVB )
σ(0) - σNaCl [VB]0
(5)
With these parameters experimentally determined ΣNa and ΣCP were found assuming that at very low VB composition there is no condensation (φ ) 1) and at very high VB composition there is full condensation (φ ) 0.36). With all the above parameters thus determined, φ(t) was computed from σ(t) by
φ(t) )
σ(t) - σNaCl - [VB]ΣVB ([VB]0 - [VB])(ΣNa + Finst,VBΣCP)
(6)
The result is depicted in Figure 6. The graph is a composite containing the behavior of φ from most of the experiments in Table 1, since each experiment covers φ only over a certain range. The fit to φ(Finst,VB) is according to an empirical sigmoidal function given by
φ(Finst,VB) ) 0.27 + 0.73/{1 + (1.969Finst,VB)2.843}
(7)
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Subsequently, the charge density was computed from φ(t) by
ξ(Finst,VB) )
lBφ(Finst,VB)Finst,VB b
(8)
where lB is the Bjerrum length, which is 0.718 nm in pure water at 25 °C, b is the contour length of both VB and Am in a polymer chain, equal to 0.26 nm. Also shown in Figure 6 is ξ computed by eqs 7 and 8. A second approach could take into account that the specific conductivity due to the copolyelectrolyte ΣCP, changes with the significant composition drift that occurs in the VB/Am system, vis-a`-vis the low drift Q9/Am system of ref 1. The contribution to conductivity due solely to the copolyelectrolyte is σcp and is the product of the charge density of VB in copolyelectrolyte form, which is ([VB]o - [VB])φ, times the mobility of the copolyelectrolyte chain, which is Ze/f, where Z is the total (noncondensed) charge due to VB in the copolyelectrolyte, and f ) 6πηR is the frictional factor, where R is the hydrodynamic radius of the chain. In the electrophoretic-free draining limit, suggested by many reports of the molecular weight independence of mobility,1-3 f R N and Z ) Finst,VBφN, where N is the degree of polymerization, so that Ze/f R Finst,VBφ. In fact, if ΣCP,0 is taken as the conductivity of a homopolymer VB chain without any counterion condensation, then the conductivity slopes dσ/ d[VB] can be expressed in this model as
dσ ) ΣVB - ΣNaφ - Finst,VBΣcp,0φ2 d[VB]
Figure 7. Average composition and linear charge distributions, zVB and zξ, respectively, for reaction 11.
zξ and zVB are scaled by lB/b () 2.8), so that, if there were no counterion condensation then zξ and zVB would overlap. It is also noted that reaction 11 had two phases. The second phase, in which neutral pAm is produced after the VB is exhausted would yield a high spiked function at ξ ) 0. This spike is omitted on Figure 7 for clarity.
(9) Summary
This further emphasizes that there is progressive counterion condensation and that φ decreases with Finst,vb. If φ remained constant then dσ/d[VB] would decrease, which contradicts the primary, model-independent data in Figure 13. The average molar composition distribution in VB is defined as
zVB )
dftotal,molar dFinst,VB
(10)
Likewise, the average charge distribution zξ can be defined as
zξ )
{(
dftotal,molar dftotal,molar lB dφ φ + Finst,VB ) dξ dFinst,VB b dFinst,VB
)}
-1
(11)
Clearly, if φ ) constant, then zξ R zVB; that is, the average composition and charge density distributions are proportional. Because of gradual counterion condensation, however, φ * constant (eq 7), and so the proportionality no longer holds. Figures 7 (reaction 11) is an example of the average composition and linear charge density distributions, using φ from eq 6. The salient features are that at low compositions of VB (low Finst,VB) the zξ and zVB overlap, since there is no appreciable counterion condensation, then grow apart as Finst,VB increases, and, finally, for high Finst,VB the chains have a nearly fixed high value of ξ, due to strong counterion condensation. This latter feature is seen in the sharp buildup of zξ around ξ ) 1, whereas zVB continues at an approximately constant value up to 0.65. The x-scales for
Online monitoring of copolyelectrolyte synthesis has been utilized to gain a comprehensive chemical and physical description of the process, including comonomer conversion, evolution of Mw, and [η]w. Direct evidence of variable fractional counterion condensation was seen by the nonlinearity of σ vs [VB] in all high-composition drift experiments, whereas it was linear in low-drift experiments. By invoking models with certain limiting features the amount of counterion condensation could be estimated by combining the conductivity and composition data. It was then possible within individual experiments to monitor the changing fraction of counterions condensed as the composition of the copolymer drifted during synthesis. This leads to a generalized form of the uncondensed fraction of counterions φ(Finst,VB) for this comonomer pair, which allows estimation of the linear charge density and the average linear charge density distribution. Acknowledgment. The authors acknowledge support from NSF CBET 0623531, Louisiana Board of Regents ITRS RDB-5, the Tulane Institute for Macromolecular Engineering and Science, NASA NNX08AP04A, and the Tulane Center for Polymer Reaction Monitoring and Characterization (PolyRMC). References and Notes (1) Garcia, G. G.; Kreft, T.; Alb, A. M.; de la Cal, J. C.; Asu´a, J. M.; Reed, W. F. J. Phys. Chem. B 2008, 112, 14597. (2) Hermans, J. J.; Fujita, H. Proc. Akad. Amsterdam 1955, B58, 182. (3) Hoagland, D. A.; Arvanititdou, E.; Welch, C. Macromolecules 1999, 32, 6180. (4) Foerster, S.; Schmidt, M. AdV. Polym. Sci. 1995, 120, 51. (5) Yamakawa, H., Modern Theory of Polymer Solutions; Harper and Row: New York, 1972. (6) Odijk, T. Biopolymers 1979, 18, 3111.
Regime Crossover under High-Composition Drift (7) Nierlich, M.; Williams, C. E.; Boue, F.; Cotton, J. P.; Daoud, M.; Farnoux, B.; Jannink, G.; Picot, C.; Moan, M.; Wolff, C.; Rinaudo, M.; de Gennes, P. G. J. Phys. (Paris) 1979, 40, 701. (8) Nallet, F.; Jannink, G.; Hayter, J.; Oberthur, R.; Picot, C. J. Phys. (Paris) 1983, 44, 87. (9) Maier, E. E.; Krause, R.; Deggelmann, M.; Hagenbuechle, M.; Weber, R.; Fraden, S. Macromolecules 1992, 25, 1125. (10) Wang, L.; Bloomfield, V. A. Macromolecules 1991, 24, 5791. (11) Drifford, M.; Dalbiez, J. P. J. Phys. Chem. 1984, 88, 5368. (12) Li, X.; Reed, W. F. J. Chem. Phys. 1991, 94, 4568. (13) Forster, S.; Schmidt, M.; Antonietti, M. Polymer 1990, 31, 781. (14) Morfin, I.; Reed, W. F.; Rinaudo, M.; Borsali, R. J. Phys. (Paris) 1994, 4, 1001. (15) Reed, W. F. J. Chem. Phys. 1994, 100, 7825. (16) Booth, F. Proc. R. Soc. London, Ser. A 1950, 203, 533. (17) Basu, S. Nature 1951, 168, 341. (18) Manning, G. S. J. Chem. Phys. 1969, 51, 934. (19) Oosawa, F. Polyelectrolytes; Marcel Dekker: New York, 1971. (20) Deserno, M.; Holm, C.; May, S. Macromolecules 2000, 33, 199. (21) Chu, J. C.; Mak, C. H. J. Chem. Phys. 1999, 110, 2669.
J. Phys. Chem. B, Vol. 113, No. 24, 2009 8309 (22) Wilson, R. W.; Bloomfield, V. A. Biochemistry 1979, 18, 2192. (23) Hinderberger, D.; Spiess, H. W.; Jeschke, G. J. Phys. Chem. B 2004, 108, 3698. (24) Florenzano, F. H.; Strelitzki, R.; Reed, W. F. Macromolecules 1998, 31, 7226. (25) Giz, A.; Koc, A. O.; Catalgil-giz, H.; Alb, A. M.; Reed, W. F. Macromolecules, 2002, 35, 6557. (26) Kreft, T.; Reed, W. F. Macromol. Chem. Phys. 2008, 209, 2463. (27) Norwood, D. P. R.; Reed, W. F. Int. J. Polym. Anal. Charact. 1997, 4, 99–132. (28) Alb, A. M.; Paril, A.; C¸atalgil-Giz, H.; Giz, A.; Reed, W. F. J. Phys. Chem. B 2007, 111, 8560. (29) Enohnyaket, P.; Kreft, T.; Alb, A. M.; Drenski, M. F.; Reed, W. F. Macromolecules 2007, 40, 8040. (30) Zimm, B. H. J. Chem. Phys. 1948, 16, 1093–1099. (31) Alb, A. M.; Enohnyaket, P.; Drenski, M. F.; Head, A.; Reed, A. W.; Reed, W. F. Macromolecules 2006, 39, 5705.
JP901672C
