Single Polyelectrolyte Species

Chapter III: Polyelectrolyte Phase Behavior

3.3 Single Polyelectrolyte Species

0.005 0.01 0.015 0.02 0.05

0.1 0.15

ρ

_p

b

³

ρ

s

b

3 ₁₀⁻³⁰ ₁₀⁻²⁰ ₁₀⁻¹⁰

10⁻¹⁰ 10⁻⁵ N = 1024

N= 175 N = 100

N= 32 N = 16 N= 64 N= 320 N = 640

∼s a l t f r e e

ρ_s= 0.3ρ_p

Figure 3.1: ρs vs. ρp phase diagram for single-species polyelectrolyte solution at lb= 1< l^c_b, b= f = 1. A linear boundary that is roughly∼0.3ρsb³demarcates an effectively salt free regime; below the line no phase separation happens. Increasing chain length increases the area of the two-phase region. The tie lines (blue dashed) are for N=100, and indicate a lower salt concentration in the polymer-rich phase.

with one multivalent species.[81] The tie lines predict a lower salt concentration in the dense than in the dilute phase, which is in agreement with previous work. [76]

This tie line behavior has been attributed to the larger excluded volume interactions in the dense phase [82] but in our recent work can be understood more generally as the result of asymmetries between small ions and polyelectrolytes in the exchange chemical potential. [83]

The density of the dense phase increases with increasing chain length, but appears to approach a common envelope for long chains, consistent with the picture of a semidilute solution independent of chain lengths. For the longest chains studied at lbf/b = 1, the polymer density remains low and never exceeds 0.02/b³. On the other hand, the supernatant phase continues to be rapidly depleted with increasing chain length due to the ever decreasing translational entropy of chains.

The upper critical salt concentration increases as a function of chain lengthN, again reflecting the increasing propensity for polyelectrolytes to phase separate. The upper critical polymer concentration is non-monotonic with chain length – it first

10

¹

10

²

10

³

10

⁻¹⁵

10

⁻¹⁰

10

⁻⁵

10

⁰

N

ρ b

3

N^−5.1 N^−4.2 upp er critical p oint

lower critical p oint

ρ^c_sb³ ρ^c_pb³

Figure 3.2: Lower and upper critical concentrations for both salt (ρ^c_s, blue) and polyelectrolyte (ρ^c_p, green), atlb = b = f = 1. The upper critical polyelectrolyte concentration is non-monotonic with a maximum atN ≈38. The lower critical con- centration vanishes very rapidly with increasing chain length, with aN-dependence ρ^crit_s ∼ N⁻⁴. The lower critical polymer concentration has an even stronger N- dependenceρ^crit_p ∼ N⁻⁵.

increases (forN < 38) and then decreases (forN > 38). The decrease for long chain lengths reflects the decreasing translational entropy of long chains, which results in the longer chains phase separating at lower densities. Meanwhile, the correlation energy becomes independent of N for long chains in semidilute solution and has negligible chain-length effect on phase separation. The initialincreasein the critical polymer concentration is because for short chains, increasing chain length initially increases the correlation energy faster than the translational entropy decreases. The behavior for these upper critical concentrations is shown by the two top curves in Fig. 3.2.

The lower critical salt concentration decreases with increasing chain length, consis- tent with the broadening of the phase separating region, and eventually vanishes for long chains (Fig. 3.2). In stark contrast, for chains with fixed Gaussian structure, sufficiently long chains do not have a lower critical point altogether. Increasing chain length in the fg-RPA leads to phase separation even in salt-free conditions, [50] as indicated by the “chimney” feature in the binodal of a Gaussian chain (Fig.

0.02 0.04 0.06 0.05

0.1 0.15 0.2 0.25 0.3 0.35

ρ

_p

b

³

ρ

s

b

3

10⁻²⁵ 10⁻¹⁵ 10⁻⁵ 10⁻⁸

10⁻⁶ 10⁻⁴ 10⁻²

ρpb³ ρsb3

f g- R PA

Our

T he or y R o d

fg-RPA Rod

Our Theory

Figure 3.3: Phase diagram of phase separation atlb= b= f =1 for N = 100. For comparison, the results for semiflexible rod (red) and for Gaussian chain (green) structures are also included. The solid lines are for the binodal, while the dashed lines are for the spinodal. The Gaussian chains phase separates for all salt concentrations studied.

3.3).

From Fig. 3.2, we see that the lower critical polymer concentration has a stronger N-dependence than the critical salt concentration. Although we cannot explain the scaling exponents for the lower critical concentrations, we suggest that the critical concentration and thus exponents may be insensitive to chain structure. To support this suggestion, we plot the binodal and spinodal for rods (l_eff = N b) and note that they have a similar shape to the RGF for the flexible chains, with a lower critical point nearly coinciding with that of the flexible chain (our theory). Remarkably, even the spinodal of the fg-RPA (l_eff =0.5), which features an instability that persists to zero salt concentration, has an ellipsoidal region with a lower apex near the lower critical point of the stiffer chains. The insensitivity of this apex to the chain structure can be understood by noting that the polymer concentration at the critical point ρp ∼ N⁻⁵ is far below overlap, in a regime where the polyelectrolyte predominantly screens as anN f-valent ion, independent of chain structure.[50] At higher concentrations, however, chain structure has a stronger impact on the electrostatic correlation and the spinodal of the fg-RPA deviates significantly from that of stiffer chains.

We also point out that at the conditions presented (lbf/b = 1), the TPT-1 theory doesnotpredict a loop in the phase diagram. The electrostatic correlation energies in TPT-1 theory are much weaker than in the RGF theory. For a polyelectrolyte solution where all monomers have diameter b and are charged, TPT-1 predicts a lower critical Bjerrum lengthlb/b≈2.8 below which the loop in the phase diagram ceases to exist.[76] For the parameters we investigated, we did not yet encounter the lower criticallb for which the loop vanishes, though the loop does shrink with decreasinglb(and should cease to exist forlb=0).

Lastly, in Fig. 3.3 one can clearly see that increasing stiffness reduces the two- phase region of the phase diagram. Within the RGF theory, this is largely because stiffer chains have smaller correlation energies. Stiffer chains naturally tend to conformations with lower electrostatic energy than more flexible chains, meaning there is consequently less correlation energy to be gained by phase separation. Also important to note is that fully flexible chains can behave much more similarly to rods than to chains with fixed Gaussian structure. As will be shown later, at sufficiently high (but still modest) charge fractions, flexible chains will locally stiffen and expand, meaning their charge environment will more closely resemble rods than truly ideal chains.