Manipulation of Confined Polyelectrolyte Conformations Through

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Manipulation of Confined Polyelectrolyte Conformations Through Dielectric Mismatch Trung Dac Nguyen, and Monica Olvera de la Cruz ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.9b03900 • Publication Date (Web): 12 Aug 2019 Downloaded from pubs.acs.org on August 12, 2019

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Manipulation of Confined Polyelectrolyte Conformations Through Dielectric Mismatch Trung Dac Nguyen† and Monica Olvera de la Cruz∗,‡,† †Department of Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208 ‡Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208 E-mail: [email protected] Abstract We demonstrate that a highly charged polyelectrolyte confined in a spherical cavity undergoes reversible transformations between amorphous conformations and a four-fold symmetry morphology as a function of dielectric mismatch between the media inside and outside the cavity. Surface polarization due to dielectric mismatch exhibits an extra “confinement” effect, which is most pronounced within a certain range of the cavity radius and the electrostatic strength between the monomers and counterions and multivalent counterions. For cavities with a charged surface, surface polarization leads to an increased amount of counterions adsorbed in the outer side, further compressing the confined polyelectrolyte into a four-fold symmetry morphology. The equilibrium conformation of the chain is dependent upon several key factors including the relative permittivities of the media inside and outside the cavity, multivalent counterion concentration, cavity radius relative to the chain length, and interface charge density. Our findings offer insights into the effects of dielectric mismatch in packaging and delivery of polyelectrolytes across media with different relative permittivities. Moreover, the reversible transformation of the polyelectrolyte conformations in response to environmental permittivity allows for potential applications in biosensing and medical monitoring.

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Keywords: confined polelectrolytes, spherical confinement, four-fold symmetry, dielectric mismatch, polarization effects, coarse-grained simulations

The thermodynamic and structural behaviors of a highly charged polymer confined into a volume comparable to its size have been of interest both practically and fundamentally. In gene delivery applications, 1,2 for example, DNA and RNA molecules are packaged inside protein capsids and lipid membranes. In living cells, proteins and DNAs are organized into micrometer-sized liquid droplets (also known as membraneless organelles), whose interior resembles an organic solvent with a lower relative permittivity than the cytoplasm. 3,4 In such conditions, the difference in the relative permittivity between the media inside and outside the confinement space leads to polarization effects, which manifest themselves as induced charges at the interface between the two media. An insightful understanding of the collective effects of spatial confinement and dielectric mismatch on the polyelectrolyte conformational behavior is therefore crucial for efficient packaging, stabilization and transfer of polyelectrolytes across media with different dielectric constants. Furthermore, if the confined polyelectrolyte conformations can be manipulated through changes in the relative permittivity of the outside medium, one can envision that the polyelectrolyte-containing complex could be used as a component of stimuli-responsive nanodevices. The influence of spatial confinement on polyelectrolytes has been extensively investigated. 5–15 The equilibrium conformation of a highly charged polymer chain confined in a cavity is evidently governed by the interplay between energetic and entropic factors coming from the confinement shape and size relative to the chain length, the chain bending stiffness, the electrostatic interaction between the charged monomers, counterions and the solvent, as well as their configurational entropy. For short flexible polyelectrolytes confined in a neutral spherical cavity, Kumar and Muthukumar 9 using self-consistent field theory demonstrated that, for a given radius of the spherical cavity and fixed charge density along the backbone of the chain, solvent and small ion entropies dominate over all other contributions to the free energy. As the chain length increases, the contribution of chain conformational entropy and polymer-solvent interaction energy to the free energy become more pronounced. The effects of confinement on the chain conformation and 2


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The jump in R∗g when surface charge density q f change from zero to nonzero values (Figure 2B) corresponds to the change from amorphous to the four-fold symmetry conformations. Compared to the amorphous collapsed conformations, the four-fold symmetry conformations have larger R∗g because the charged monomers spread out into the two perpendicular planes. For q f > 0, however, the equilibrium radius of gyration R∗g decreases monotonously with the increasing surface charge density q f (Figure 2B and Supporting Information, Figures S10 and S11). This is because the polyelectrolyte in the four-fold morphology is compressed further towards the cavity center. This trend is only valid when q f is sufficiently low so to keep the confined polyelectrolyte within the conformational entropy-dominated regime, as already pointed out by Wang and Muthukumar. 10 Since the four-fold symmetry conformations are not spherical, the radius of gyration R∗g is certainly not the ideal order parameter, but can serve as a reasonable metric for comparing the size of the morphologies obtained from different nonzero surface charge densities. The changes in the slopes of the free energy landscape and in the equilibrium radius of gyration with respect to q f can be explained as follows. When dielectric mismatch is sufficiently large, the amount of the outer counterions that are drawn toward, and adsorbed on, the interface from outside is increased. Consequently, the surface induced charge on the interface is amplified in the presence of polarization effects (Figure 2C). The competition between the repulsion from the increased number of adsorbed outer negatively charged counterions and the attraction from the positively charged interface gives rise to the four-fold symmetry morphology of the polyelectrolyte.

Conformational free energy profiles as a function of confinement volume, multivalent counterion fraction and electrostatic strength inside the cavity In this section we focus on the influences of several key parameters on the free energy profile of the chain squared radius of gyration: confinement volume, multivalent counterion fraction, and electrostatic strength inside the cavity, with surface charge density assumed to be zero (q f = 0). In this case, surface polarization effects due to dielectric mismatch alone simply shift R∗g to a slightly smaller value (Supporting Information, Figure S12). 9


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Confinement volume To examine the effects of spatial confinement on the chain equilibrium conformations, we vary the cavity radius, R, keeping the chain length, N, unchanged. As can be seen from Figure 3, for a cavity radius as small as R = 5σ , the free energy almost monotonously increases with Rg , indicating that the chain is highly compressed by the spherical interface. This is the regime where spatial confinement is dominant and completely shadows surface polarization. As R increases, the free energy minimum R∗g shifts to greater values, and the effect of spatial confinement is negligible for R = 7σ . It is interesting to note the noticeable change in the free energy profile when R is increased from 5σ to 6σ , with the equilibrium size R∗g expanded along with the reduced free energy gradient for Rg > R∗g . Likewise, when the chain length is increased for the same cavity radius, R∗g /Nσ 2 shifts to smaller values when the confinement degree increases (Supporting Information, Figure S13). Because the cavity size R = 6σ indicates the regime where spatial confinement effects come into play, yet not too strong, we will focus on this value of the cavity radius in the following sections. 50

R = 5σ R = 6σ R = 7σ

40 A(R2g)/kBT

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Figure 3: Effects of confinement size, R, on the free energy profile in the presence of surface polarization for φm = 0.3. The relative permittivity outside the cavity is kept at εout = 4. Interface charge density is q f = 0.

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Multivalent counterion fraction The conformational features of polyelectrolytes in salt-free solution 28–30 and in the presence of monovalent and multivalent counterions 31–39 have been thoroughly investigated. The precipitation of polyelectrolytes and the random coil-collapse transition of polyelectrolytes in a dilute solution induced by monovalent and multivalent counterions can be explained by thermodynamic models 31–35 and computer simulations. 5,36–38 González-Mozuelos and Olvera de la Cruz 30 developed a thermodynamic model using mean field approximation and short-range correlation corrections to investigate the equilibrium conformations of highly charged polyelectrolytes in salt-free dilution solutions. They demonstrated that the ratio between the counterion and monomer valencies is among the key factors that determine whether the collapsed or the rod-like conformations have lower free energy. Frutos et al. demonstrated that the tetravalent cation spermine, as a DNA condensing agent, reduces the pressure inside the capsid and influences the DNA ejection from the bateriophage. 35 It is evident that, at sufficiently high counterion concentration, the Coulombic attraction between the counterions and charged monomers is responsible for the random coil-collapse transition of the polyelectrolyte. The size of the collapsed chain is determined by the chain bulkiness and the counterion valency and concentration. This picture is indeed corroborated in our results. The effects of the multivalent counterion fraction, φm , on the conformational behavior of the polyelectrolyte are shown in Figure 4. While the change in the slope of the free energy profile is rather small, the collapsed radius R∗g decreases monotonously from 3.7σ to 3.3σ as φm increases from 0 to 0.3. This is because all the multivalent counterions are condensed on the polyelectrolyte and bridge different chain segments (Supporting Information, Figure S14), and because trivalent counterions experience stronger repulsion from the interface due to surface polarization than monovalent counterions do. Whereas, the fraction of monovalent counterions condensed on the polyelectrolyte decreases as φm increases. The system minimizes its potential energy with trivalent counterions-monomer Coulombic attractions, and simultaneously gains additional translational entropy of the released monovalent counterions. 11


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φm = 0.3 φm = 0.2 φm = 0.1 φm = 0.0

A(R2g)/kBT

8 6 4 2 0

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12 R2g/σ2

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Figure 4: Effects of multivalent counterion fraction, φm , on the free energy profile. The cavity radius is R = 6σ . Interface charge density is q f = 0. 50

εin = 10 εin = 20 εin = 40

40 A(R2g)/kBT

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Figure 5: Effects of the electrostatic strength inside the cavity on the free energy profile in the presence of surface polarization for φm = 0.3. The relative permittivity outside the cavity is kept at εout = 4. Interface charge density is q f = 0. Electrostatic strength inside the cavity We show in Figure 5 the effects of the Bjerrum length inside the cavity lB ∼ 1/εin on the free energy profile by varying εin . As εin decreases, lB , and hence the strength of the Coulombic interaction between the charged particles, increases. On the one hand, the amount of counterions condensed on the polyelectrolyte increases. On the other hand, the repulsive forces between likecharged particles also become stronger. The combined effects of the Coulombic attraction and surface polarization, which pushes the charged particles away from the interface, then lead to the noticeable increase in the work needed to unravel the chain as εin decreases, hence the increased 12


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gradient of the free energy profile for Rg > R∗g . Whereas for Rg < R∗g , decreasing εin lowers the polyelectrolyte free energy due to the energetic gain associated with the increased Coulombic attraction between the monomers and counterions. These behaviors are in accord with the modified “ion-bridging” model developed for highly charged polyelectrolytes at high concentrations 31,32 and with the regime where the counterion concentration inside the cavity is sufficiently low to favor the collapse of the polyelectrolyte. 37

Relaxation time of the Rouse modes of the polyelectrolyte The influence of the counterions on polyelectrolyte dynamics is certainly nontrivial, at the length scales of the monomers and of the sub-chains. 40,41 Webb and coworkers showed that the dynamics of a single negatively charged polyelectrolyte is substantially suppressed across multiple length scales for different concentrations of lithium cations compared to that of neat polymers. 41 Here we also characterize the dynamics of the confined polyelectrolyte at different length scales through the relaxation time of its Rouse modes. The Rouse modes of a polyelectrolyte chain of length N 42 The are defined as X p (t) = (2/N)1/2 ∑N i=1 ri (t) cos[(p/N)(i − 1/2)], where p = 0, 1, 2, ..., N − 1.

smallest mode (p = 0) captures the motion of the chain center of mass, whereas the other modes describe the motion of the segments composed of (N − 1)/p monomers. We compute the normalized autocorrelation function of the Rouse modes hX p (t)·X p (0)i/hX p (0)· X p (0)i, where h·i is the average over multiple time origins. Figure 6 shows the log plot of the autocorrelation function of several representative Rouse modes for p ≤ 20, corresponding to the sufficiently long sub-chains to be consistent with the Rouse model. Up to a certain time interval, the autocorrelation function fits well with the simple exponentially decaying function e−t/τ p , where τ p is defined as the relaxation time of mode p. τ p decreases as p increases, as indicated by the steeper slopes in the log plot. To evaluate the influence of the Coulombic interaction between the charged monomers and counterions on the dynamics of the polyelectrolyte, we compute the relaxation time of the Rouse modes of the chain with the Coulombic interaction switched off, that is, the non-bonded interaction 13


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between the monomers and counterions is only the WCA potential. As seen in the inset of Figure 6, τ p /τ pn > 1 for all p, meaning that the relaxation times of the Rouse modes of the polyelectrolyte are always longer than those of a neutral chain. Particularly, the relaxation time at the length scale of the whole chain (p = 1) is greater than that of the neutral counterpart by an order of magnitude for all the values of φm studied. As p increases, the relaxation time of the sub-chains at shorter length scales approaches, but is still longer than, that of the neutral chain. The longer correlation time in the dynamics of the polyelectrolyte across multiple length scales is attributed to the ionic correlation between the charged monomers and the counterions, which in consistent with the findings reported in the recent work by Webb and coworkers on ion-doped polymers. 41 We note on passing that polarization effects alone have negligible effects on the relaxation time of the Rouse modes, at least for this particular case. 1.0

p=1 p=5

0.5 p=8

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Figure 6: Normalized autocorrelation function of the Rouse modes of the confined polyelectrolyte in a neutral cavity (q f = 0). The trivalent counterion fraction is φm = 0.3. Inset shows the relaxation time for several Rouse modes, τ p , normalized by the corresponding relaxation time of a neutral chain of the same length N, τ pn , for different multivalent counterion fractions, φm . The cavity radius is R = 6σ .

CONCLUSIONS We have demonstrated that a highly charged polyelectrolyte confined in a spherical cavity adopts two distinct morphologies, that is, the amorphous and four-fold symmetry conformations, depending on dielectric mismatch between the media inside and outside the cavity and surface charge 14


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density. The transition between the two morphologies is reversible upon changes in the relative permittivity of the outside medium. Surface polarization due to dielectric mismatch exhibits an extra “confinement” effect, which is most pronounced within a certain range of the cavity radius and the electrostatic strength between the monomers and counterions and multivalent counterions. Interestingly, for cavities with a charged surface, surface polarization leads to an increased amount of counterions adsorbed in the outer side, further compressing the confined polyelectrolyte into the four-fold symmetry conformations. The equilibrium conformation of the chain is dependent upon several key factors including multivalent counterion concentration, cavity radius relative to the chain length, and the relative permittivity of the medium inside the cavity. Our findings offer insights into the effects of dielectric mismatch in packaging and delivery of polyelectrolytes across media with different relative permittivities. Furthermore, it suggests that dielectric mismatch can be realized as a control knob to manipulate the confined polyelectrolyte morphologies, and vice versa, that the confined polyelectrolyte morphologies can be used to monitor the change in the relative permittivity of the outside medium.

METHODS We develop a model system of a linear polyelectrolyte with explicit counterions confined in a spherical cavity. Example physical systems of interest include single-stranded DNAs, RNAs and macromolecules that are highly charged. In these systems, the interior medium of the cavity typically contains water molecules, which are necessary to dissolve the polyelectrolyte and counterions. While the dielectric response of the interior medium is not necessarily uniform, it is reasonable to the first order of approximation that it is in the range of 10–40, i.e., the value at the surface of many proteins. 43 The exterior of the cavity is composed of thick layers of amphiphilic molecules with hydrophilic head groups sticking inward to interact with the essential water molecules, charged monomers and counterions, and hydrophobic groups pointing outward. The coarse-grained model of the confined charged polyelectrolyte system is shown in Figure

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7. The polymer chain is represented by a bead-spring model 24 composed of N = 100 coarsegrained monomers, each approximately 5-10 Å in diameter. As such, the bead diameter maps roughly to one Kuhn length. We add Nc+ monovalent counterions and Nc3+ trivalent counterions to neutralize the total charge of the polymer: Nc+ + 3Nc3+ = N. The spherical interface has a radius of R, discretized into M vertices (or patches). The polymer beads and counterions interact with the spherical confinement wall via the Weeks–Chandler–Andersen (WCA) potential. 25 The multivalent counterion fraction is defined as φm = N 3+ /N. We use the same value of σ = 1.0 for the WCA interaction between all the counterions to represent their hydrodynamic diameters. For charged interfaces (q f > 0), negatively charged counterions are added in the outer medium to ensure charge neutrality. In our model, we assume that the counterions do not diffuse through the cavity shell, which is relevant for the cases where the shell thickness is sufficiently thick to make the rate of ion transport extremely low, and/or when the Gibbs-Donnan equilibrium of the counterions inside and outside the cavity has been reached. The outer counterion concentration is varied from 0.004σ −3 − 0.02σ −3 , corresponding to 20–100 mM. The thickness of the cavity shell, δ , is varied between 1.0 − 2.0σ by setting an outer spherical wall with radius Rout = R + δ that interacts with the outer counterions via the WCA potential. The electrostatic interaction strength inside the cavity is characterized by the Bjerrum length, √ lB = q2 σ /εin , where q = Ze/ 4πε0 σ kB T is the dimensionless charge of the monovalent counterions (Z = 1). Unless otherwise stated, the relative permittivity of the medium inside and outside the cavity are εin = 40 and εout = 4, respectively. For εin = 40, the monomer charges q are set so that lB /σ = 4, reminiscent of single stranded DNAs and NaPSS. The polyelectrolyte and counterions are equilibrated in the canonical ensemble (constant volume and temperature) using the Nose-Hoover chain thermostat at kB T /ε = 1, where ε = 1.0 is the well depth of the WCA potenp tial. The dimensionless time unit is τ = σ m/ε, where m = 1 is the bead mass, which is the same for all the particles. The integration timestep is ∆t = 0.001τ. To characterize the conformational behavior of the confined polyelectrolyte, we perform umbrella sampling simulations using SSAGES, a software suite for accelerated sampling, 44 coupled

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Coarse-grained model

e beads (-q) N = 100 counterions (+q) rions (+3q) es m counterions nderson + Coulombic for nonand FENE bonds Angstroms (e.g. ssRNA)

nts in/out = 40/4 side the cavity: q2 lB = = 4s ein 4pe0 ein kB T

!out < !in

– –

–

+

!in

+

– –

+3

– –

+3

+ – –

–

– –

– –

+

–

AAACCHicbVBNSgMxGM34W+tf1aWbYBFclRkR1F3RjcsKji20Q8mkmTY0fyYZsQw9gQdwq0dwJW69hSfwGmbaWdjWB4HHe99fXqwYNdb3v72l5ZXVtfXSRnlza3tnt7K3f29kqjEJsWRSt2JkCKOChJZaRlpKE8RjRprx8Dr3m49EGyrFnR0pEnHUFzShGFknRR2iDGVSdDMqxt1K1a/5E8BFEhSkCgo0upWfTk/ilBNhMUPGtANf2ShD2lLMyLjcSQ1RCA9Rn7QdFYgTE2WTo8fw2Ck9mEjtnrBwov7tyBA3ZsRjV8mRHZh5Lxf/89qpTS4i9x+VWiLwdFGSMmglzBOAPaoJtmzkCMKaulshHiCNsHU5zWzJZyvLn/JkgvkcFkl4WrusBbdn1fpVEVEJHIIjcAICcA7q4AY0QAgweAAv4BW8ec/eu/fhfU5Ll7yi5wDMwPv6BUhtm7I=

qf ≥ 0

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

on fraction:

fm = N AAACDnicbVBNSsNAGJ34W+tfqks3g0UQhJqooC6EohtXpYK1hTaGyXTSDp1JwsxELSGH8ABu9QiuxK1X8ARew0mbhW198MHjve+P50WMSmVZ38bc/MLi0nJhpbi6tr6xaZa27mQYC0waOGShaHlIEkYD0lBUMdKKBEHcY6TpDa4yv/lAhKRhcKuGEXE46gXUpxgpLblmqRP1qcvhBazdJ8cH6WHNNctWxRoBzhI7J2WQo+6aP51uiGNOAoUZkrJtW5FyEiQUxYykxU4sSYTwAPVIW9MAcSKdZPR6Cve00oV+KHQFCo7UvxMJ4lIOuac7OVJ9Oe1l4n9eO1b+mZPQIIoVCfD4kB8zqEKY5QC7VBCs2FAThAXVv0LcRwJhpdOauJLtjhR/SnUy9nQOs6RxVDmv2Dcn5eplHlEB7IBdsA9scAqq4BrUQQNg8AhewCt4M56Nd+PD+By3zhn5zDaYgPH1CzOUm+w=

3+

eout < ein AAACG3icbVDLSsNAFJ3UV62vqEtBBovgqiQiqOCi6MZlBWMLbSiT6aQdOnkwcyOWkJ3f4Qe41U9wJW5d+AX+hpM2oG09MHA45z7mHi8WXIFlfRmlhcWl5ZXyamVtfWNzy9zeuVNRIilzaCQi2fKIYoKHzAEOgrViyUjgCdb0hle537xnUvEovIVRzNyA9EPuc0pAS11zv8NixYWmaZRAhi/wr8DDrGtWrZo1Bp4ndkGqqECja353ehFNAhYCFUSptm3F4KZEAqeCZZVOolhM6JD0WVvTkARMuen4jgwfaqWH/UjqFwIeq387UhIoNQo8XRkQGKhZLxf/89oJ+GeuvidOgIV0sshPBIYI56HgHpeMghhpQqjk+q+YDogkFHR0U1vy2TEED3ky9mwO88Q5rp3X7JuTav2yiKiM9tABOkI2OkV1dI0ayEEUPaJn9IJejSfjzXg3PialJaPo2UVTMD5/ACbXow4=

Dielectric interface discretized into a counterions and trivalent coun/N Figure 7: Simulation setup for a polyelectrolyte with monovalent 7

0 in atriangular f M= 642–2652 vertices terionsq confined spherical mesh cavity. ofThe media inside and outside the cavity are assumed to be uniform with relative permittivities, εin and εout , respectively. The dash lines represent the cavity thickness where counterions and charged monomers are excluded. AAACBHicbVDLTgIxFO3gC/GFunTTSExckRljou6IblxiIoKBCemUDjS0nbG9YyQT1n6AW/0EV8at/+EX+Bt2YBYCnqTJyTn31RPEghtw3W+nsLS8srpWXC9tbG5t75R39+5MlGjKGjQSkW4FxDDBFWsAB8FasWZEBoI1g+FV5jcfmTY8UrcwipkvSV/xkFMCVrp/6Ia402fY7ZYrbtWdAC8SLycVlKPeLf90ehFNJFNABTGm7bkx+CnRwKlg41InMSwmdEj6rG2pIpIZP50cPMZHVunhMNL2KcAT9W9HSqQxIxnYSklgYOa9TPzPaycQnvspV3ECTNHpojARGCKc/R73uGYUxMgSQjW3t2I6IJpQsBnNbMlmxyCfxjYZbz6HRdI4qV5UvZvTSu0yj6iIDtAhOkYeOkM1dI3qqIEokugFvaI359l5dz6cz2lpwcl79tEMnK9fTl+Y1A==

with LAMMPS. 45 Essentially, SSAGES employs LAMMPS for sampling the polyelectrolyte conformation in the canonical ensemble, and at every time step applies the biasing forces to the charged monomers based on the difference between the instantaneous value of the collective variable, R2g in this case, and the constrained value set within each window, R2g0 . The biasing potential applied to the monomers at every time step is given by U (b) = 0.5k(R2g − R2g0 )2 , where k is the biasing (b)

constant, which gives the biasing force on individual monomers: fi

= −∇ri U (b) . We tested with

k = 50, 75 and 100 (in unit of ε/σ 4 ), and chose k = 75 for all of the simulations reported to ensure that the histograms of R2g obtained from the adjacent windows highly overlap.

We use 32–48 windows for umbrella sampling, each corresponding to a constrained value of the polyelectrolyte squared radius of gyration, R2g0 . The polarization solver in use is the induced charge computation (ICC∗ ) method 46 implemented in LAMMPS by our previous work. 47 The biased histograms Pb (R2g ) obtained from individual windows are then combined together using the weighted histogram analysis method 48 to yield the free energy profile. Note that all the free energy curves are shifted to zero at their minimum in the WHAM calculations. In the present study, we 17

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have performed more than 500 replica simulations in total, each equilibrated for 5000-10000τ. By fitting the autocorrelation function of the system potential energy with exp(−t/τr ), we found that the relaxation time τr is found to be on the order of the time unit τ for the simulated systems. There are more than 1000 simulations (as replicas and as independent runs) in total performed in the present study.

ASSOCIATED CONTENT Supporting Information Supporting Information Available: Simulation results for various parameter sets that lead to amorphous and four-fold symmetry conformations of the confined polyelectrolyte. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author ∗ E-mail:

[email protected].

ORCID Trung Dac Nguyen: 0000-0002-5076-264X Monica Olvera de la Cruz: 0000-0002-9802-3627

ACKNOWLEDGEMENTS The research was supported by the Sherman Fairchild Foundation and by the Center for Computation and Theory of Soft Materials. We thank Michael Engel for sharing the code used to generate the diffraction patterns.

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Graphical TOC Entry

Dielectric constants in/out = 40/4 Bjerrum length inside the cavity: q2 lB = = 4s ein 4pe0 ein kB T

!out < !in

–

–

+

!in

+

– +3

–

– –

+3

+ – –

–

–

– –

– –

+

–

AAACCHicbVBNSgMxGM34W+tf1aWbYBFclRkR1F3RjcsKji20Q8mkmTY0fyYZsQw9gQdwq0dwJW69hSfwGmbaWdjWB4HHe99fXqwYNdb3v72l5ZXVtfXSRnlza3tnt7K3f29kqjEJsWRSt2JkCKOChJZaRlpKE8RjRprx8Dr3m49EGyrFnR0pEnHUFzShGFknRR2iDGVSdDMqxt1K1a/5E8BFEhSkCgo0upWfTk/ilBNhMUPGtANf2ShD2lLMyLjcSQ1RCA9Rn7QdFYgTE2WTo8fw2Ck9mEjtnrBwov7tyBA3ZsRjV8mRHZh5Lxf/89qpTS4i9x+VWiLwdFGSMmglzBOAPaoJtmzkCMKaulshHiCNsHU5zWzJZyvLn/JkgvkcFkl4WrusBbdn1fpVEVEJHIIjcAICcA7q4AY0QAgweAAv4BW8ec/eu/fhfU5Ll7yi5wDMwPv6BUhtm7I=

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

Trivalent counterion fraction:

fm = N 3+ /N AAACDnicbVBNSsNAGJ34W+tfqks3g0UQhJqooC6EohtXpYK1hTaGyXTSDp1JwsxELSGH8ABu9QiuxK1X8ARew0mbhW198MHjve+P50WMSmVZ38bc/MLi0nJhpbi6tr6xaZa27mQYC0waOGShaHlIEkYD0lBUMdKKBEHcY6TpDa4yv/lAhKRhcKuGEXE46gXUpxgpLblmqRP1qcvhBazdJ8cH6WHNNctWxRoBzhI7J2WQo+6aP51uiGNOAoUZkrJtW5FyEiQUxYykxU4sSYTwAPVIW9MAcSKdZPR6Cve00oV+KHQFCo7UvxMJ4lIOuac7OVJ9Oe1l4n9eO1b+mZPQIIoVCfD4kB8zqEKY5QC7VBCs2FAThAXVv0LcRwJhpdOauJLtjhR/SnUy9nQOs6RxVDmv2Dcn5eplHlEB7IBdsA9scAqq4BrUQQNg8AhewCt4M56Nd+PD+By3zhn5zDaYgPH1CzOUm+w=

eout < ein

qf ≥ 0

AAACG3icbVDLSsNAFJ3UV62vqEtBBovgqiQiqOCi6MZlBWMLbSiT6aQdOnkwcyOWkJ3f4Qe41U9wJW5d+AX+hpM2oG09MHA45z7mHi8WXIFlfRmlhcWl5ZXyamVtfWNzy9zeuVNRIilzaCQi2fKIYoKHzAEOgrViyUjgCdb0hle537xnUvEovIVRzNyA9EPuc0pAS11zv8NixYWmaZRAhi/wr8DDrGtWrZo1Bp4ndkGqqECja353ehFNAhYCFUSptm3F4KZEAqeCZZVOolhM6JD0WVvTkARMuen4jgwfaqWH/UjqFwIeq387UhIoNQo8XRkQGKhZLxf/89oJ+GeuvidOgIV0sshPBIYI56HgHpeMghhpQqjk+q+YDogkFHR0U1vy2TEED3ky9mwO88Q5rp3X7JuTav2yiKiM9tABOkI2OkV1dI0ayEEUPaJn9IJejSfjzXg3PialJaPo2UVTMD5/ACbXow4=

qf AAACBHicbVDLTgIxFO3gC/GFunTTSExckRljou6IblxiIoKBCemUDjS0nbG9YyQT1n6AW/0EV8at/+EX+Bt2YBYCnqTJyTn31RPEghtw3W+nsLS8srpWXC9tbG5t75R39+5MlGjKGjQSkW4FxDDBFWsAB8FasWZEBoI1g+FV5jcfmTY8UrcwipkvSV/xkFMCVrp/6Ia402fY7ZYrbtWdAC8SLycVlKPeLf90ehFNJFNABTGm7bkx+CnRwKlg41InMSwmdEj6rG2pIpIZP50cPMZHVunhMNL2KcAT9W9HSqQxIxnYSklgYOa9TPzPaycQnvspV3ECTNHpojARGCKc/R73uGYUxMgSQjW3t2I6IJpQsBnNbMlmxyCfxjYZbz6HRdI4qV5UvZvTSu0yj6iIDtAhOkYeOkM1dI3qqIEokugFvaI359l5dz6cz2lpwcl79tEMnK9fTl+Y1A==

0

Dielectric interface discretized into a triangular mesh of M = 642–2652 vertices

1.6 1.6 1.4 1.2 1.2 1.0 0.8 0.8 0.6 0.4 0.4 0.2 0.0 0.0

Δε/− ε Dielectric mismatch

Coarse-grained model Cyan: Polyelectrolyte beads (-q) N = 100 Yellow: Monovalent counterions (+q) Red: Trivalent counterions (+3q) Gray: Interface charges Green: Outer medium counterions - Weeks-Chandler-Anderson + Coulombic for nonbonded interactions; and FENE bonds - Unit length σ ~ 10 Angstroms (e.g. ssRNA) - Implicit solvent

7

Four-fold symmetry

Amorphous

00 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 Surface charge density 2

qfσ /q

24

Manipulation of Confined Polyelectrolyte Conformations Through

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