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## New Challenges in Electrostatics of Soft and Disordered Matter

#### University of Toulouse III, France

#### Organisers

-- The workshop is now closed for applications --

Recently there has been a surge of activity in order to elucidate the behavior of highly charged soft matter and Coulomb fluids in general. Such systems are ubiquitous especially in biological matter where the length-scale and strength of the interaction between highly charged biomolecules are governed by strong electrostatic effects. The theoretical starting point to describe the highly charged and correlated matter is the strong coupling theory, which is apart and distinct from the well-known Poisson-Boltzmann weak-coupling regime. The strong coupling theory has been successfully applied to explain the observation of the multivalent ion mediated attraction between like-charged objects. Recently, several interesting limits have been discovered in the parameter space of highly charged many-particle Coulomb matter where analytical progress is possible and leads to completely novel and unexpected results [1]. One of the challenges in highly charged matter is to correctly describe systems with finite coupling strength, in the transition regime between the weak and strong coupling. Studying the fluctuations about both, the WC and the SC limits, several theories have attempted to describe this experimentally most relevant regime. At the same time, computer simulation algorithms and the computing power have advanced to the level where all-ion simulations including many-body and polarization effects are possible and the new theories can be a subject to numerical confirmation. Another important question is the effect of the structural disorder on electrostatic interactions. Recently it has recently been demonstrated that charge disorder can impose a long-range interaction between charged or even uncharged surfaces and such interactions might be turn out to be of key relevance in biological processes.

The primary purpose of the proposed workshop is to cross-pollinate the different theoretical and simulation approaches and to develop a single perspective that would tie together the low temperature, high coupling constant and disorder parameters in a unified description of the electrostatic interactions, which, to a large extent, determine the stability and conformations of most important biological macromolecules.

Recently there has been a surge of activity in order to elucidate the behavior of highly charged soft matter and Coulomb fluids in general. Such systems are ubiquitous especially in biological matter where the length-scale and strength of the interaction between highly charged biomolecules are governed by strong electrostatic effects. The theoretical starting point to describe the highly charged and correlated matter is the strong coupling theory, which is apart and distinct from the well-known Poisson-Boltzmann weak-coupling regime. The strong coupling theory has been successfully applied to explain the observation of the multivalent ion mediated attraction between like-charged objects. Recently, several interesting limits have been discovered in the parameter space of highly charged many-particle Coulomb matter where analytical progress is possible and leads to completely novel and unexpected results [1]. One of the challenges in highly charged matter is to correctly describe systems with finite coupling strength, in the transition regime between the weak and strong coupling. Studying the fluctuations about both, the WC and the SC limits, several theories have attempted to describe this experimentally most relevant regime. At the same time, computer simulation algorithms and the computing power have advanced to the level where all-ion simulations including many-body and polarization effects are possible and the new theories can be a subject to numerical confirmation. Another important question is the effect of the structural disorder on electrostatic interactions. Recently it has recently been demonstrated that charge disorder can impose a long-range interaction between charged or even uncharged surfaces and such interactions might be turn out to be of key relevance in biological processes.

The primary purpose of the proposed workshop is to cross-pollinate the different theoretical and simulation approaches and to develop a single perspective that would tie together the low temperature, high coupling constant and disorder parameters in a unified description of the electrostatic interactions, which, to a large extent, determine the stability and conformations of most important biological macromolecules.

Electric charges are ubiquitous in soft and biological systems as most soft materials such as polymers, colloids, and proteins acquire surface charges when they dissolve in water and release small mobile ions into the bathing solution. These materials are easily deformed by potentials comparable to thermal energy and thus electrostatic forces constitute a prominent factor determining the structure and properties of these materials in various applications [1]. In industry, charged macromolecules are used, due to their high water-solubility, in a multitude of applications such as in design and processing of non-toxic environmentally friendly materials. In biology, where systems of this type are particularly important as they set the length-scale and strength of the interaction between highly charged biomolecules, electrostatic effects emerge in many examples such as in the interactions of the cell membranes and macromolecules, transport through the membrane channels, DNA packaging in the cell nucleus as well as in bacteriophages. The latter deploy tightly packed DNA as an osmotic coil whose energy is used to drive the infection process, as well as in the bulk where DNA condensation induced by polyvalent cations leads to tightly packed toroidal aggregates with a diameter of up to a few hundred nanometers [1]. It has now been an established fact for a while that in these cases electrostatic forces enter in a counter-intuitive fashion leading to strong attraction between like-charged segments of DNA. Recent studies have shown that such phenomena are dominated by attractive counterions-induced forces between juxtaposed macromolecular surfaces. They emerge when the surfaces are highly charged and counterions are multivalent [1,2,3,4,5].

Several fundamentally different approaches have been proposed to understand the behavior of highly charged systems. One of them is the low temperature Wigner crystal ansatz that starts with the zero temperature ground state [2]. In this limit the interaction between two equally charged macroscopic surfaces is attractive due to the crystallization of counterions in the 2D Wigner lattice close to the surfaces, that is correlated across the space between them. Several attempts have been made to look also at the fluctuations about this ground state [6]. The second approach starts with the weak coupling Poisson-Boltzmann limit. The fluctuations about the mean-field limit have been studied to a much larger extent and have been shown to generalize the concept of the temperature-dependent Casimir interactions [7]. There are interesting cases where thermal fluctuations of this type, which are usually deemed to be sub-dominant to the leading order mean-field contribution, give the most important part of the total interaction free energy between electrified interfaces. This can lead to an effective fluctuation-induced attraction between like-charged surfaces. Last but not least, there is systematic expansion of the partition function in terms of the strength of the electrostatic interactions that leads to the strong coupling theory as formulated by Netz, based on the functional-integral representation of the partition function of Coulomb fluids [4,5]. This approach can be viewed as a limiting law strictly valid for highly charged Coulomb fluids. In this case, too, the interaction between two opposed equally charged surfaces can be attractive but for different reasons to the aforementioned cases, stemming mostly from the bridging configurations of the highly charged counterions between the surfaces. Certain aspects of realistic Coulomb systems are shared by exactly soluble 1d and 2d Coulomb systems, where there have been a number of notable recent advances. Such idealized systems help our understanding of various approximations and help us to assess their applicability in well controlled situations. The mathematical formalism introduced by Edwards and Lenard [8] to study the bulk properties of the one dimensional Coulomb gas has been used to study the interaction between charged surfaces in 1d [9]. Also, recent developments in integrable systems open up the possibility of examining the same problems in two dimensional Coulomb gases [10]. While all these approaches are built around the charge as the only descriptor of ions and surfaces, there are indications that ion-specific effects [14] and solvent mediated interactions [15] are just as important in determining the range and sign of interactions in highly charged Coulomb systems [16].

Apart from the strength of the Coulomb interactions in realistic systems one also has to deal with structural disorder present ubiquitously. One manifestation of the structural disorder is the disordered nature of the fixed charge distribution at the bounding surfaces [11]. Relaxing the ansatz of a uniform surface charge density and allowing for a disordered component of this distribution has led to emergence of distinct phenomena leading to extremely long-range interaction potentials between disordered charge distributions of this type [12]. The fundamental interest in these disorder-driven interactions notwithstanding, they are clearly important in the quantitative analysis of the high-accuracy experiments of Casimir and related interactions that have recently witnessed a fast-paced progress mostly due to the introduction of novel experimental methods for measuring directly the interaction at the nano-scale [13].

While there are cases where a physical system is in a regime where one of the approximation schemes alluded to above may be applicable, in most realistic systems there are multiple length scales which means that the system cannot be understood within a single regime. Computer simulations are presently practically the only tool available to gain a reliable insight into the behavior of the systems in between the limiting regimes accessible to theories. A workshop which assembles experts using various theoretical and simulational approaches is timely in order to discuss how the theories can be linked with each other in order to describe complex systems containing a number of regimes and finally compare the predictions with the experiments.

## References

**China**

Jure Dobnikar (Chinese Academy of Sciences) - Organiser

Rudolf Podgornik (University of Chinese Academy of Sciences, Beijing) - Organiser

**France**

David Dean (University Paul Sabatier, Toulouse) - Organiser

**Iran**

Ali Naji (Institute for Research in Fundamental Sciences, Tehran) - Organiser