## Abstract

We analyse theoretically and by means of molecular dynamics (MD) simulations the generation of mechanical force by a polyelectrolyte (PE) chain grafted to a plane. The PE is exposed to an external electric field that favours its adsorption on the plane. The free end of the chain is linked to a deformable target body. By varying the field, one can alter the length of the non-adsorbed part of the chain. This entails variation of the deformation of the target body and hence variation of the force arising in the body. Our theoretical predictions for the generated force are in very good agreement with the MD data. Using the theory developed for the generated force, we study the effectiveness of possible PE-based nano-vices, composed of two clenching planes connected by PEs and exposed to an external electric field. We exploit the Cundall–Strack solid friction model to describe the friction between a particle and the clenching planes. We compute the diffusion coefficient of a clenched particle and show that it drastically decreases even in weak applied fields. This demonstrates the efficacy of the PE-based nano-vices, which may be a possible alternative to the existing nanotube nano-tweezers and optical tweezers.

This article is part of the themed issue ‘Multiscale modelling at the physics–chemistry–biology interface’.

## 1. Introduction

Future nanotechnology will use molecular devices executing various manipulations with nano-size objects, such as colloidal particles, vesicles, macromolecules, viruses, small bacteria, cell organelles, etc. Such objects, however, cannot stay at rest on their own due to heat, manifesting itself in the form of thermal fluctuations. To execute highly precise manipulations with these objects, one needs to keep them immobilized. Therefore, devices that can clench and unclench nano-size objects under external control will be in great demand. These devices, which can be termed ‘nano-vices’ or ‘nano-nippers’, should be able to operate in solutions, including aqueous solutions, where manipulations with biological nano-size objects are expected. Recently, nanotube nano-tweezers operated by an electric field, which can clench a nano-size object, have been proposed [1]. Such devices, however, cannot work effectively in aqueous solutions; moreover, the range of forces, as well as the range of operating distances, is relatively narrow for such devices [1]. Optical tweezers require specific optical properties of an immobilized object, which also restricts their applications [2]. Therefore, it seems reasonable to consider polyelectrolyte (PE)-based nano-vices, operated by an electric field, which could be a very promising alternative to existing nanotube nano-tweezers and optical tweezers. These can operate in aqueous solutions and demonstrate a wide range of operating forces and distances. They also overcome limitations imposed by the optical properties of nano-objects [2].

In very simple terms, the nano-vices may be composed of a few charged polymer chains, the so-called PEs, that are linked to two surfaces and exposed to an electric field that serves as a control signal. By varying the field, one can alter the length of the non-adsorbed part of the chains and hence the distance between the planes (figure 1*a*). In this way, one can clench and unclench a nano-size object (target body) placed between the planes. Recent technological achievements, e.g. the production of nano-sheets [3], enable a practical realization of such devices; hence it is very important to develop a theory of the respective devices and quantitatively describe the basic physical processes there. Hence one needs (i) to develop a theory of a conformational response of a PE, with one end linked (grafted) to a charged surface and the other one to a deformable target body, and exposed to a varying electric field, and (ii) to quantify the ability of nano-vices to clench a nano-size object.

The response of PEs to external electric fields has been extensively studied in the last few decades [4–12], including the adsorption of PEs on oppositely charged surfaces of different geometries [13,14]. In particular, the conformational response of a PE in an electric field, under the action of a constant force, has been analysed [5]. In the context of nano-vices, however, this problem has been addressed only recently [10–12]. Moreover, the quantification of the clenching ability of PE-based nano-vices has not been studied yet. In [10], a theory of the phenomenon, based on a model ‘physical’ approach, has been developed for the case of a constant force acting on a free (non-grafted) end of a PE. Later, in [11], a generalization of this theory for the case of force depending on deformation has been reported. In [12], a first-principles theory of this phenomenon has been elaborated; all the theoretical studies were accompanied by extensive molecular dynamics (MD) simulations [10–12].

In this paper, we first discuss the theoretical approach of [10–12] and give a shorter and more straightforward derivation of the main result of the first-principles theory [12]. Then we analyse the efficacy of nano-vices by calculating the diffusion coefficient of a clenched particle. In particular, we show that for rather weak electric fields the PE-based nano-vices can effectively immobilize a clenched Brownian particle, reducing its diffusion coefficient by a few orders of magnitude.

## 2. Conformation of a polyelectrolyte linked to a target body in an external electric field

### (a) Model

We consider a system composed of a chain of *N*_{0}+1 monomers, which is anchored to a planar surface at *z*=0. The anchoring end-monomer is uncharged. Each of the remaining *N*_{0} beads carries the charge −*qe* (*e*>0 is the elementary charge) and *N*_{0} counter-ions of charge +*qe* make the system neutral. For simplicity, we consider a salt-free solution. However, it may be shown [15] that, for intermediate and strong electric fields, the presence of salt ions (up to physiological concentrations) leads to a simple renormalization of the external field, so that the qualitative nature of the phenomenon remains unchanged.^{1} Hence a salt-free case, which allows an analytical treatment, is generic.

We use the freely jointed chain model, with the length of the inter-monomer link equal to *b*. The MD simulations [10–12] justify the application of this model to the analysed phenomena. We consider a salt-free system, so that no other micro-ions are present in the solution, which has the dielectric permittivity *ε*. The chain is grafted (linked) to a charged plane so that the external electric field **E** acts perpendicular to the plane, favouring the chain adsorption. The free end of the PE is linked to a deformable body, modelled by a spring (figure 1*a*(ii)).

The deformation energy of the spring *U*_{sp}=*U*_{sp}(*h*−*h*_{0}) depends on the deformation *h*−*h*_{0}, where *h*_{0} and *h* are the sizes of the undeformed and deformed target body, respectively. The restoring force acting on the free end of the chain then reads
2.1In [10–12], the following dependences have been studied in detail theoretically and numerically: *U*_{sp}=*fh*, , *U*_{sp}=(*κ*/*γ*)|*h*−*h*_{0}|^{γ−1} and , which refer respectively to the constant force, linear, nonlinear and Hertzian spring. Here *κ* is the elastic constant of the spring and *θ*(*x*) is the Heaviside step function.

For the values of the external field and system parameters addressed in this study, the counter-ions are practically decoupled from the chain, being accumulated near the upper plane (figure 1*a*(iii)). Hence the interaction of the chain with the external field as well as interactions between the chain monomers are not screened by the counter-ions. To find the conformational response of the PE to the external electric field and the force arising in the target body, one needs to find the free energy of the system and minimize it with respect to relevant parameters.

### (b) Free energy of the chain

We first calculate a conditional free energy of the chain *F*_{ch} with the following conditions imposed: (i) *N* monomers of the chain are desorbed and *N*_{s}=*N*_{0}−*N* are adsorbed on the charged plane; (ii) the distance between the free chain end, linked to the target body, and the grafting plane is *z*_{top}; and (iii) the end-to-end vector of the adsorbed part of the chain is **R**. Minimizing the conditional free energy *F*_{ch}(*N*,*z*_{top},**R**) with respect to these variables, one can find the equilibrium values of *N*, *z*_{top} and **R**; using these quantities we then compute the force acting on the target body.

For the freely jointed model the locations of all monomers of the chain are determined by *N*_{0} vectors **b**_{i}=**r**_{i}−**r**_{i+1}, which join the centres of (*i*+1)th and *i*th monomers; each of these vectors has the same length *b*. We enumerate the monomers starting from the free end linked to the target body. Then the beads with numbers 1,2,…,*N* refer to the bulk part of the chain, while those with numbers *N*+1,*N*+2,…,*N*_{0} refer to the surface part. The (*N*_{0}+1)th neutral bead, located at the origin, **r**_{N0+1}=0, is anchored to the surface. We assume that the adsorbed part of the chain forms a flat structure, so that the centres of the adsorbed beads lie in the plane *z*=0. In other words, we ignore the off-surface loops of this part of the chain. Then the location of the *k*th bead of the chain and the distance between the centres of *i*th and *j*th beads read
2.2The orientation of each vector **b**_{s} is characterized by the polar *θ*_{s} and azimuthal *ψ*_{s} angles, where the axis *OZ* is directed perpendicular to the grafting plane (figure 1*a*(iii)). Therefore, the distance between the grafting plane at *z*=0 and the *k*th bead of the bulk part of the chain as well as the height of the top bead for *k*=1 read as
2.3The location of the top bead, linked to the target body, determines the deformation energy of the target body, and the corresponding force which acts on the chain is
2.4where *z*_{top, 0} is the coordinate of the top bead for the case when the target body is not deformed. As the chain is not screened by the counter-ions, the potential associated with the external field *E* is *φ*_{ext}(*z*)=−*Ez*. Hence the interaction energy of the bulk part of the chain with the external potential reads
The interaction energy with the external field of the adsorbed part of the chain, located at *z*=0, is equal to a constant, which we take equal to zero.

The electrostatic interactions between the chain monomers are also unscreened and may be written using the Fourier transform of a Coulomb potential *q*^{2}*e*^{2}/*εr* as [12]
2.5where we exploit equation (2.2). Taking into account that the end-to-end vector of the adsorbed part of the chain has the form , we can finally write the conditional free energy as
2.6where *β*=(*k*_{B}*T*)^{−1} with *k*_{B} and *T* being, respectively, the Boltzmann constant and temperature, and is the conditional partition function:
2.7where the factor *b*^{3} keeps dimensionless and *ψ*_{i} and *θ*_{i} are, respectively, the azimuthal and polar angles of an inter-monomer vector **b**_{i} (figure 1*a*(iii)). We also take into account that *θ*_{N+1}=*θ*_{N+2}=⋯=*θ*_{N0}=*π*/2, since the adsorbed part of the chain forms a flat structure. Note that **R** is a two-dimensional vector on the plane and the vectors **b**_{i}, for *i*=*N*+1,…,*N*_{0}, have zero *z* component.

Using the integral representation for the delta function,
where the vector **r** has a lateral and *z* components, **r**=(**r**_{⊥},*z*), we recast equation (2.7) into the form
2.8where and we define
2.9Here we take into account that , that *N*_{s}=*N*_{0}−*N*, and use the integral representation of the zeroth-order Bessel function, . We also define the averaging over the azimuthal angles *ψ*_{s} as
2.10Taking into account the long-range nature of the Coulomb interactions in *H*_{self}, one can expect that the mean-field approximation will have a good accuracy. The mean-field approximation deals with the average quantities and neglects fluctuations. The average quantities are described by the first-order cumulants, while fluctuations are described by the higher-order cumulants. Hence we adopt the following mean-field approximation:
2.11Performing integrations over *ψ*_{1},…,*ψ*_{N0} and then over **k**, we arrive at (see e.g. [12] for detail, where similar quantities have been computed):
2.12where *l*_{B}=e^{2}/(*εk*_{B}*T*) is the Bjerrum length, and . Next, integration in equation (2.8) over **p**_{⊥} yields
2.13where we use from equation (2.9) together with the approximation that , which is valid due to the large exponent *N*_{s}≫1 in equation (2.9). To evaluate the integral in equation (2.13), we also apply the steepest descent method with the saddle point at .

Finally, we integrate over *η*_{1},…,*η*_{N} in equation (2.8), applying the mean-field approximation
and using again the steepest descend method in the integration over *p*_{z}. This eventually leads to the following expression for the conditional free energy:
2.14where and is the saddle point. We also define
2.15with Ei(*x*) being the exponential integral function, and , and the contribution of the electrostatic interaction energy between the bulk and adsorbed parts of the PE chain [12]:
2.16where .

The impact of counter-ions on the conformation of the bulk part of the chain may be estimated as a weak perturbation. Referring for the computational detail to [12], we give here the final result:
2.17where *μ*_{GC}=1/(2*πσ*_{c}*l*_{B}*q*) is the Gouy–Chapman length based on the apparent surface charge density *σ*_{c}=*qN*_{0}/*S* associated with the counter-ions and *S* is the lateral area of the system.

### (c) Dependence of the force and deformation on the external field

Now we can determine the dependence on the electric field of the PE dimensions as well as the deformation of the target body. Simultaneously, one obtains the dependence on applied field of the force that arises between the chain and the target body. This may be done by minimizing the total free energy of the system *F*(*N*,*z*_{top},*R*)=*F*_{ch}(*N*,*z*_{top},*R*)+*F*_{count}(*z*_{top}) with respect to *N*, *z*_{top} and *R* and using *N*_{s}=*N*_{0}−*N* and the constraint *z*_{top}≤*bN*. This allows one to find *N*, *z*_{top} and *R* as functions of the applied electric field, that is, to obtain *N*=*N*(*E*), *z*_{top}=*z*_{top}(*E*) and *R*=*R*(*E*). Then one can compute the force acting onto the target body. It reads
2.18where , with *f*(*z*_{top})=−∂*U*_{sp}/∂*z*_{top} being the reduced force for a particular force–deformation relation. In the above equation, we exploit and the saddle point equation, i*z*_{top}−∂*W*(*p*_{z})/∂*p*_{z}=0, valid for *p*_{z}=*p**.

## 3. Clenching efficiency of nano-vices

To quantify the efficiency of clenching by nano-vices, we analyse diffusion of a particle squeezed by the planes of the device. One can say that the body is effectively kept at rest if the diffusion coefficient of the clenched particle drops down by a few orders of magnitude, when compared with this value for a free particle.

For simplicity, we assume that the two planes are kept parallel and that two equal forces, normal to the plane, act on the top and bottom of the particle. We neglect twisting and rolling motion of the clenched particle and consider only sliding, that is, the tangential motion. When the particle moves tangentially, it experiences the following set of forces. (i) A viscous force from the surrounding fluid, **F**_{vis}=−*γ***v**, acts against the velocity **v** with the friction coefficient *γ*=6*πηR*_{p}, where *η* is the fluid viscosity and *R*_{p} is the radius of the particle. (ii) A random force acts from the surrounding fluid, ** ξ**(

*t*), related to the viscous friction. We assume that this is a

*δ*-correlated force that obeys the fluctuation–dissipation theorem: (iii) A ‘solid friction’ force acts between the planes and the particle,

**F**

_{sol.fr}. Hence one can write the stochastic equation of motion for the Brownian particle between the planes: 3.1

The microscopic derivation of the force **F**_{sol.fr} is rather challenging; therefore we exploit here the Coulomb friction model in the microscopic interpretation of Cundall & Strack [16]. Note that this model has been used for the tangential friction between colloidal particles [17]. In the Cundall–Strack model, it is assumed that the solid friction force, which counteracts an externally applied force, is equal to a harmonic spring force −*κ*Δ, where Δ=(**r**−**r**_{0}) and **r**_{0} is the initial position of the body, before the external force started to act. After the displacement from the initial position reaches some quantity *d*, the spring breaks down and the body remains displaced by Δ=*d* in the direction of the acting force. For the next action of the external force, the new initial position **r**_{0} corresponds to the shifted one. The maximal force in this model is *κd*, while the average force is equal to . On the macroscopic scale *L*≫*d*, the body moves smoothly with the average resistance force equal to *F*_{sol.fr}=*μF*_{⊥}=*κd*/2, where *μ* is the friction coefficient, which yields, *κ*=2*μF*_{⊥}/*d*. One can also write the solid friction force, in the regime of the acting spring, as the derivative of the corresponding potential, *F*_{sol.fr}=−∇*U* with *U*=*μF*_{⊥}(**r**−**r**_{0})^{2}/*d*. Since the solid friction force always acts against the applied force, we can write the above equation (3.1) as a one-dimensional equation in the direction of the applied stochastic force ** ξ**. Moreover, if we assume that the viscous force is large, , we recast the above Langevin equation into the overdamped form:
3.2with , where

*D*

_{0}=

*k*

_{B}

*T*/

*γ*is the diffusion coefficient of a non-constrained particle.

Note that, in contrast to the conventional overdamped Langevin equation, the above equation (3.2) describes the potential *U*(*x*), centred after each jerk of size *d*, at a new position. Hence the random tangential motion of a particle occurs as follows. If a displacement of the particle due to the action of the stochastic force *ξ*(*t*) does not exceed *d*, the particle performs a Brownian motion in the harmonic potential *U*(*x*), centred at **r**_{0}, with zero diffusion coefficient. If at some moment *t*_{0} the displacement becomes equal to **d**, the particle shifts to **r**=**r**_{0}+**d** and starts to perform Brownian motion in the potential *U*(*x*), centred at **r**=**r**_{0}+**d**. Let the critical displacements by vectors **d**_{0}, **d**_{1},…, **d**_{k} occur at times *t*_{0},*t*_{1},…,*t*_{k}. Owing to the Markovian properties of the random force *ξ*(*t*), all the vectors **d**_{k} and the instances *t*_{k} are independent. Moreover, we can assume that the sequence of times *t*_{1},…, *t*_{k} obeys the Poisson distribution
where *τ* is the average time between the successive jerks. Then the mean square displacement during time *t* reads
3.3which implies that the diffusion coefficient is *D*=*d*^{2}/2*τ*. In the above equation, we take into account that 〈**d**_{i}⋅**d**_{j}〉=*δ*_{ij} *d*^{2}, due to independence of the displacements **d**_{i} and **d**_{j} for *i*≠*j*.

Hence, to find the diffusion coefficient, one needs to compute the average waiting time *τ*. This may be done if we find the so-called ‘mean first-passage time’ for the potential *U*(*x*) [18,19]. The equation for the average time *T*(*x*) needed for a particle, initially located at a point *x*, within a potential well *U*(*x*), to reach the point *x*=±*d*, where the jerk takes place, reads [18]
3.4where *U*(*x*)=*μF*_{⊥}*x*^{2}/*d*=*α*(*x*^{2}/*d*^{2}), with *α*=*μF*_{⊥}*d*/*k*_{B}*T* and we choose **r**_{0}=0. The boundary conditions at *x*=±*d* are obvious. The solution to equation (3.4) may be expressed in terms of the hypergeometric function *H*(*z*)=_{2}*F*_{2}(1,1;3/2,2;*z*):
3.5Now we average *T*(*x*) over the starting point *x*, using the equilibrium distribution of starting points within the potential well, ∼e^{−βU(x)}. The result then reads

where is the normalization constant and in the last expression we expand *H*(*x*) around *x*=0 and keep only the leading term. If the potential well is deep enough, that is, *βU*(*d*)≫1, then the above expression may be approximated by its asymptotic value for *α*≫1. Taking into account that for *x*≫1 and using equation (3.3), we find the effective self-diffusion coefficient:
3.6Here *D* depends on the field *E* since *F*_{⊥}=*F*_{⊥}(*E*)=2*f*(*z*_{top}), which has been computed in the previous section. Using the value of *μ*=0.2, which is motivated by the friction coefficients between polymer surfaces (e.g. *μ*=0.5 for polystyrene–polystyrene, *μ*=0.15–0.25 for nylon–nylon, etc.), and taking into account that the friction force acts on the top and bottom parts of a clenched particle, which duplicates the friction force, we calculate *D* as a function of the electric field. The results for *D*(*E*) are shown in figure 2*b*. For simplicity in our calculation, we use *d*=*b* for the microscopic parameter of the Cundall–Strack friction model.

## 4. Molecular dynamics simulations

The MD simulations have been performed for a freely jointed bead–spring chain. All *N*_{0}=320 beads, except the one bound to the plane at *z*=0, carry one (negative) elementary charge. *N*_{0} monovalent free counter-ions of opposite charge make the system electroneutral. We assume an implicit good solvent, which implies short-range, purely repulsive interactions between all particles, described by a shifted Lennard-Jones potential. Neighbouring beads along the chain are connected by the standard FENE potential (e.g. [20,21]). The bond length at zero force is *b*≃*σ*_{LJ} with *σ*_{LJ} being the Lennard-Jones parameter. All particles except the anchor bead interact with a short-range repulsive potential with the grafting plane at *z*=0 and upper plane at *z*=*L*_{z}. The charged particles interact with each other and the external field with unscreened Coulomb potential, quantified by the Bjerrum length. We set *l*_{B}=*σ*_{LJ} and use a Langevin thermostat to hold the temperature *k*_{B}*T*/*ϵ*_{LJ}=1, where *ϵ*_{LJ} is the Lennard-Jones energy parameter. More simulation details are given in [20,21]. The free end of the chain is linked to a deformable target body, which is modelled by springs with various force–deformation relationships. We use the Hertzian force, which can describe the ‘core’ interactions between nanoparticles [22]. For simplicity, we assume that the anchor of a spring is fixed and aligned in the direction of the applied field. The footprint of the simulation box is *L*_{x}×*L*_{y}=424×424 (in units of *σ*_{LJ}) and the box height is *L*_{z}=*L*=160. A typical simulation snapshot is shown in figure 1*a*(iii). We observed that already for relatively weak fields, starting from a field of about *Eqeb*/*k*_{B}*T*≈0.1 and higher fields, the adsorbed part of the chain is almost flat, while the bulk part of the PE chain in strongly stretched along the field. Moreover, in sharp contrast to the field-free case [23–25], the counter-ions are decoupled from the PE and accumulate near the upper plane.

## 5. Results and discussion

In figure 2*a*, we show the results of MD simulations and compare them to the theoretical predictions. In particular, the dependence of the field-induced force acting on the target body is shown as a function of the external electric field. As may be seen from the figure, very good agreement between the theory and MD data is observed in the absence of any fitting parameters. Note, however, that the theory has been developed for a highly charged chain with a relatively strong self-interaction and interactions with the charged plane. Some systematic deviations are observed for large fields for the largest nanoparticle. This may probably be attributed to a possible elongation of the bond length for strong forces, which is not accounted for in the freely jointed chain model with the fixed bond length *b*. Note that, for aqueous solutions at ambient conditions, the characteristic units of the force and field are *k*_{B}*T*/*b*≈*k*_{B}*T*/*l*_{B}≈6 pN and *k*_{B}*T*/*be*≈*k*_{B}*T*/*l*_{B}*e*≈35 V μm^{−1}, respectively; the latter value is about an order of magnitude smaller than the critical breakdown field for water. Also note that the range of electric fields roughly corresponds to a surface charge density of 0.1–10 C cm^{−2}, typical for charged graphite surfaces [26,27].

In figure 2*b*, the dependence of the diffusion coefficient of the clenched particle on the external field is shown. One can see a dramatic decrease of *D* even for relatively weak fields. The effect becomes even more pronounced for larger particles, where the characteristic field of reduces the diffusion coefficient by nine orders of magnitude. This proves the effectiveness of the PE-based nano-vices.

## 6. Conclusion

We investigate the generation of a mechanical force by a PE chain grafted to a plane and exposed to an external electric field, when it is linked to a deformable target body. MD simulations are performed and an analytical theory of this phenomenon is elaborated. The focus is on the case when the force–deformation relationship corresponds to that of a Hertzian spring, which quantifies repulsive interactions between nano-size objects, like colloidal particles. The theoretical dependences for the generated force, acting on the target body, on the external field are in very good agreement with the simulation data.

Based on the developed theory of the force-to-field response, we analyse the efficacy of PE-based nano-vices, composed of two planes connected by PEs and exposed to an external electric field. By varying the electric field, one can clench and unclench a nano-size particle placed between the planes. We apply the Cundall–Strack solid friction model to describe the friction between a particle and clenching planes and develop a theory for the diffusion coefficient of a clenched particle. It is shown that the diffusion coefficient of a clenched particle drastically decreases even at relatively small electric fields. This proves that PE-based nano-vices may be effectively used to clench nano-size objects as a possible alternative to the existing nanotube nano-tweezers and optical tweezers. Such devices may find a wide application in future nano-industry, when it is needed to keep a nano-size object immovable, say in various assembly processes. Among important advantages of the discussed nano-vices are (i) their possibility to operate in aqueous solutions, including solutions with salt, (ii) a wide range of operating forces and distances, which may be realized within a single device, and (iii) a large variety of molecular structures of PEs that may be used to produce nano-vices.

## Competing interests

We declare we have no competing interests.

## Funding

No funding has been received for this article.

## Authors' contributions

N.V.B. designed the study, participated in the derivation of the main analytical results and drafted the manuscript. Yu.A.B. participated in the derivation of the analytical results, performed numerical analysis of the derived equations and helped to draft the manuscript. C.S. carried out the MD simulations and the data analysis, provided the MD simulation snapshots and the MD data for the plots and helped to draft the manuscript. All authors gave final approval for publication.

## Footnotes

One contribution of 17 to a theme issue ‘Multiscale modelling at the physics–chemistry–biology interface’.

↵1 The salt co-ions simply screen the adsorbing plane for such fields, but do not practically screen the bulk part of the chain.

- Accepted August 1, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.