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# Numerical evidence for thermally induced monopoles

Contributed by Daan Frenkel, March 19, 2017 (sent for review January 3, 2017; reviewed by Lyderic Bocquet and Peter Palffy-Muhoray)

## Significance

Thermal gradients are ubiquitous in nature, yet relatively little is known about the forces they induce on the nanoscale. Here, we show using molecular simulations that a pair of heated/cooled colloidal particles in a dipolar solvent behaves like oppositely charged electric or magnetic monopoles, as recently suggested theoretically. In particular, we demonstrate that the field distribution induced in the solvent is in excellent agreement with the field generated by two homogeneously charged spheres in vacuum. This intriguing result advances our understanding of the complex interactions in nanoscale systems out of thermal equilibrium, opening unique possibilities for its applications in nanotechnology.

## Abstract

Electric charges are conserved. The same would be expected to hold for magnetic charges, yet magnetic monopoles have never been observed. It is therefore surprising that the laws of nonequilibrium thermodynamics, combined with Maxwell’s equations, suggest that colloidal particles heated or cooled in certain polar or paramagnetic solvents may behave as if they carry an electric/magnetic charge. Here, we present numerical simulations that show that the field distribution around a pair of such heated/cooled colloidal particles agrees quantitatively with the theoretical predictions for a pair of oppositely charged electric or magnetic monopoles. However, in other respects, the nonequilibrium colloidal particles do not behave as monopoles: They cannot be moved by a homogeneous applied field. The numerical evidence for the monopole-like fields around heated/cooled colloidal particles is crucial because the experimental and numerical determination of forces between such colloidal particles would be complicated by the presence of other effects, such as thermophoresis.

The existence of quasi-monopoles in a system of heated or cooled colloidal particles in a polar or paramagnetic fluid follows directly from nonequilibrium thermodynamics, combined with the equations of electro/magneto-statics (1). Although suggested theoretically, they have thus far not been studied experimentally. This paper provides numerical evidence indicating that the predicted effects are real and robust. In what follows, we consider the case of thermally induced quasi-monopoles in a dipolar liquid, but all our results also apply to paramagnetic liquids. It has been shown that a thermal gradient will create an electric field in a liquid of dipolar molecules with sufficiently low symmetry (2, 3). In the absence of any external electric field, a heated or cooled colloidal particle placed in such a liquid will generate an electric field according to the phenomenological relation (2, 4, 5)

Let us next consider the electric polarization around a heated (or cooled) colloidal particle, for brevity also referred to simply as a colloid. We note that the sole function of the colloid is to generate a temperature gradient field in the solvent, which in turn couples to the electric field via Eq. **1**. Other heat sources (sinks) would lead to the same effect. In steady state the temperature profile at a distance

## Results

To verify the existence of thermally induced charges numerically, we performed nonequilibrium molecular dynamics (NEMD) and equilibrium MD simulations of a heated and a cooled colloid immersed into a modified (“off-center”) Stockmayer fluid (7), consisting of particles with a point dipole and a Lennard–Jones (LJ) center displaced along the direction of the dipole moment (*Materials and Methods* and *Off-Center Stockmayer Model*). This displacement is controlled by a parameter *Comparison with On-Center Stockmayer Model*), in accordance with simulation studies on dumbbell molecules that identified shape or mass asymmetry as a requirement for this effect (3). An important property of our model fluid is that *Estimation of S_{TP}*), thereby facilitating the analysis compared with the polar models considered previously (2, 4, 6, 8⇓⇓⇓–12). The temperature gradient is sustained by continuously pumping energy into the hot colloid and removing it from the cold one such that the overall system energy is constant (13).

In our numerical simulations, we chose a geometry in which the two colloids are located on the *Materials and Methods*. To improve statistics, we computed cylindrical averages (indexed by *A* corresponds to the equilibrium (or bulk) temperature *A* and a value of *Estimation of S_{TP}*), we can use Eq.

**4**to obtain an estimate of

Fig. 1*B* shows the average dipolar orientations superimposed onto the electric field lines generated by two virtual point charges located at the centers of the colloids. To single out the thermally induced alignment from contributions already present in equilibrium, e.g., the alignment caused by surface layering of solvent molecules in the vicinity of the colloids, we measured equilibrium orientations in a separate simulation and subtracted them from the nonequilibrium result. This procedure assumes that the coupling between the molecular alignment present in equilibrium and the thermally induced one is negligible. We found this assumption to be reasonable everywhere apart from the immediate vicinity of the colloids. Therefore, we excluded the first layer of solvent molecules, i.e., all particles within a distance of

As a more quantitative test of the theory, we measured the electric field induced by the temperature gradient. To improve the statistical accuracy of our results, we average the field over planes perpendicular to the symmetry axis, such that all contributions apart from *A* and *B*. For this geometry, we obtain the analytical solution for the electric field (*Analytical Model for the Field*),**6** enables us to link the theory and NEMD simulations quantitatively. We can estimate the right-hand side of the above equation readily by sampling the instantaneous dipole orientations and performing temporal and spatial averaging for slabs perpendicular to the symmetry axis. Using Eq. **5**, we can then infer the value of **4**. Observing a good agreement for both estimates would provide strong support for the theory, because it would suggest that Gauss’s theorem can be applied to arbitrary volumes enclosing the colloids, just as if they carried real Coulomb charges. We note, however, that there is an important conceptual difference between estimating the charge using Eq. **5** and estimating it using Eq. **4**: The latter already assumes that Eq. **1** holds whereas the former validates it.

Fig. 2*C* shows the steady-state result for the spatial variation of the averaged field calculated according to Eq. **6**. Equilibrium averages were subtracted and solvent particles within a distance of **5**: The average field is constant in the fluid region and changes linearly within a distance of

## Discussion

A key question is whether the effective electric or magnetic charge of colloidal monopoles can be measured in experiments. The present simulations suggest that at the very least the effect of the monopole fields on probe charges (or dipoles) should be observable. Of course, it would be attractive to make the effect as large as possible by increasing the temperature difference between the particle and the solvent. However, the temperature range is limited by the fact that extreme heating or cooling will bring the system out of the linear-response regime—and possibly even induce phase transitions in the solvent. Moreover, the colloidal monopoles differ in an important respect from true monopoles: They cannot be moved by a uniform external field (1). For an isolated thermal monopole, this follows from the fact that neither its self-energy, i.e., the energy of the electric field around it, nor the interaction energy between the external field and solvent dipoles exhibits a dependence on the position of the thermal monopole. It is therefore tempting (be it slightly frivolous) to call such colloidal monopoles “quacks,” as they quack like a duck (i.e., they create a field similar to that of a real monopole), but they do not swim like a duck (they cannot be used to transport charge). One of the main effects that may obscure observation of the Coulomb-like interaction between oppositely heated colloids is thermophoresis, which will also cause colloids to move in the temperature gradient caused by another colloid. However, at least in the linear regime, this effect should cause otherwise identical but oppositely heated colloids to move in the same direction with respect to the fluid rather than with respect to one another. Finally, there are many open questions about the practical consequences of the existence of thermal monopoles. It is, for instance, conceivable that such particles in an electrolyte solution will get “decorated” with real charges and thereby acquire real charge (opposite and equal to the “thermal” charge) that can be dragged along. That charge should respond to a uniform external field: The resulting electro-osmotic flow would cause motion of the colloids.

## Materials and Methods

### Simulation Setup.

All equilibrium and nonequilibrium MD simulations were performed using the software package LAMMPS (15) (version 14Jun16). We used a fully periodic rectangular simulation box with dimensions *Off-Center Stockmayer Model*. We used the relations

### Equilibration.

The initial lattice structure was equilibrated in a simulation with a fixed number of particles *NVT* run, which was followed by a *NVE* simulation to subtract nonvanishing equilibrium averages of the spatially averaged field and the dipolar orientations from the NEMD results. The relative increase in the total energy throughout the entire NEMD production run (75 million time steps) was ∼

### Statistical Analysis.

The size of each error bar in Fig. 2*C* represents twice the standard deviation of the mean value that was calculated as the difference between the nonequilibrium and the equilibrium averages. For the individual production run we computed field averages according to the following protocol: At regular time intervals of **6**, excluding dipoles within a distance of **5**, was computed from the slabs with index **4** is omitted because we do not have error estimates for the temperature contour lines shown in Fig. 1*A*. The computation of *Estimation of S_{TP}*.

## Analytical Model for the Field

In this section we derive the analytical model proposed in Eq. **5**. To this end, we first show how the spatial average of the 3D field, _{TP} used in the main text is dropped for notational convenience. We consider periodic boundary conditions (PBCs) and understand that this is implicitly taken into account whenever an expression of the form

For an arbitrary charge distribution, the field can be calculated as

Next, we work out the averaged charge density and compute the field from Eq. **S2d**. The colloids are modeled by two homogeneously charged, spherical shells of radius **S2d** and carry out the integration, we obtain the final result

The quantity

We note that the value of **S10**. Equivalently, we can think of the result as the sum of two equal contributions, half from the charge

## Off-Center Stockmayer Model

Displacing the LJ center from the location of the point dipole leads to modified forces and torques compared with the original Stockmayer model (7). We note that electrostatic contributions are not affected by this modification and refer to ref. 21 for the relevant expressions. All modifications of short-ranged interactions related to the perturbation of the LJ center are governed by a single parameter

Let us consider the short-ranged, pairwise interactions between two solvent particles as illustrated in Fig. S2. The point dipoles of mass

The radially symmetric, pairwise LJ potential is given by

Taking the negative gradient of the energy with respect to

## Estimation of *S*_{TP}

We estimated the thermo-polarization coefficient, using the relation*C*), implying that

## Comparison with On-Center Stockmayer Model

A microscopic theory that accounts for the alignment of off-center Stockmayer particles in a thermal gradient is at present lacking, but recent studies on dumbbell molecules suggest that a shape or mass asymmetry is required for the effect (3, 5, 10). In our model shape asymmetry is introduced by choosing a nonzero value for

## Comparison of Temperature and Electric Potential

The results shown in Fig. 1 suggest that the electric field lines are aligned perpendicular to the temperature isosurfaces, implying that the electric field is parallel to the temperature gradient field. A direct quantitative comparison of the 3D fields is difficult for the following reasons: First, statistical fluctuations in the computed electric field are relatively large, which is why we considered planar averages in Fig. 2*C*. Second, computation of the temperature gradient field requires taking numerical derivatives of the simulation data, thereby increasing the error. We therefore explore an alternative route and compare the averaged electric potential, **5**), which we have already shown to be in good agreement with the simulation data (Fig. 2*C*), such that

## Estimate for Water

An accurate estimate of the thermally induced charge for water would require additional simulations with a realistic model. We can, however, get a rough idea of the order of magnitude, using an estimate of **4** to obtain an estimate of

Colloidal particles can also carry a charge due to the dissociation of ionizable groups at the surface (22). For example, polystyrene spheres of radius

## Acknowledgments

P.W. acknowledges many invaluable discussions with Martin Neumann, Chao Zhang, Michiel Sprik, Aleks Reinhardt, Carl Pölking, and Tine Curk. We acknowledge financial support from the Austrian Academy of Sciences through a doctoral (DOC) fellowship (to P.W.), the Austrian Science Fund (FWF) within the Spezialforschungsbereich Vienna Computational Materials Laboratory (Project F41) (C.D.), and the European Union Early Training Network NANOTRANS (Grant 674979 to D. Frenkel). The results presented here have been achieved in part using the Vienna Scientific Cluster.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: df246{at}cam.ac.uk.

Author contributions: P.W., C.D., and D. Frenkel designed research; P.W., D. Fijan, R.A.L., and A.Š. performed research; and P.W., A.Š., C.D., and D. Frenkel wrote the paper.

Reviewers: L.B., Ecole Normale Superieure; P.P.-M., Kent State University.

The authors declare no conflict of interest.

Data deposition: Additional data related to this publication are available at the University of Cambridge data repository (https://doi.org/10.17863/CAM.8607).

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1621494114/-/DCSupplemental.

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