| The Single Chain Mean Field Theory (SCMFT) is one of the theoretical tools exploited in the Molecular Simulation group at the Universitat Rovira i Virgili (URV). It exactly accounts for the configurations of a single microscopically detailed chain at the molecular level. The interactions between different chains are described through a mean molecular field which is found self-consistently. The methodology of the SCMFT is adapted for parallel computation via OpenMP shared memory platform. The developed code can run in parallel. The SCMFT is particularly suitable for the description of nano-objects like polymeric drug carriers: it gives a detailed microscopic information on the configurations of the chains, the optimal shape and structure of drug delivery systems, the distribution of chains in the aggregate, the critical micellar concentrations as well as the critical aggregation concentration, the optimal aggregation number and the size distributions. The method is quite universal: it can be applied to solutions of linear or branched polymers, solutions of low-molecular weight surfactants and various additives, mixtures of various components and structural and shape transitions. It can be extended for the description of ordered phases, like liquid crystals, and to disordered structures as gels.
The spirit of the SCMFT method is similar to the Density Functional Theory (DFT). The conformations of a single chain are generated via Monte Carlo simulations and the intra molecular interactions are calculated exactly. The cooperative effects of other similar chains in the solution or in the melt are found self-consistently: the chain is considered being surrounded by the fields created by over similar chains. These fields (e.g. concentrations of different components, solvent) are calculated iteratively from the self-consistency condition: individual chain conformations depend on the values of the surrounding fields, while the values of the fields are the average properties of individual conformations. After few steps of iterations the method converges to the solution which gives the equilibrium structures and the concentrations profiles in the system as well as the most probable conformations of individual chains. The power of this method is the speed in obtaining solutions (faster than Monte Carlo and much faster than Molecular Dynamics simulations). A limitation is the mean field level of description which does not take into account fluctuations. This method readily generalizes for the mixture of an arbitrary number of polymers and other objects interacting with each other in the solution. The mean field approximation is valid if there are no strong correlations between interacting particles, such as ion pair formation or bonds formation. In this case the partition function of many component systems factorizes and the free energy is written as a sum of independent terms. The free energy per volume of the multi-component system contains two terms:  In this expression ci(a,r) is the distribution function of i-component with a denoting all internal degrees of freedom and r the spatial coordinates. The first term in the sum is the translational degree of freedom of the components distinguishable by their internal degrees of freedom and the coordinates. Each of the components are assumed to consist of ni subunits which is reflected in the additional term. In case of the polymers, ni is the number of Khun segments in the chain. The second term is the interaction of the components between each other and with external fields. Each of the field ui(a,r) depends on the coordinates and internal degrees of freedom. This free energy is usually coupled with the incompressibility condition implying that the sum of concentrations of all components in a solution is fixed.
Since the objects of interest are usually composed of nano-size aggregates freely moving in a solution, the translational entropy of the aggregate as a whole can be decoupled from the entropy of moving of chains inside a single aggregate. Thus the distribution function ci(a,r) allows for factorization:
 where Ni is the number of chains inside an aggregate and the internal degrees of freedom a are decoupled from the translational coordinates of the aggregate r. With this division Xi(r)/Ni becomes the number concentration of the aggregates containing chains. Thus the free energy can be written in the form  The first term is the translational entropy of the aggregates of Ni chains. Thus the free energy and the chemical potential of the system can be written as a sum of two terms, translational entropy of the centres of mass of the aggregates and the energy and chemical potential of the aggregates themselves. We consider translational homogeneity of the solution, ui(a,r)=ui(a), but it does take into account the local inhomogeneity of the fields around the aggregates and a contains conformations of the chains and their positions from the centre of the aggregate. In the following we denote the part of the free energy and the chemical potential corresponding to internal degrees of freedom with the subscript *. The chemical potential is  And the free energy  This expression allows fixing the centre of mass of the aggregate or the lipid bilayer in the centre of coordinates and considering only the movements of the rest of the chains around the centre of mass. The translational entropy is added after the solution is found. In some cases, for example for phospholipid bilayers this term is negligibly small, since Ni is large. This division of the variables simplifies a lot the calculations: the simulation box is divided in shells according to the symmetry of the problem (spherical layers, cylindrical layers, horizontal layers or cubic lattice if there is no symmetry), The centre of the aggregate is fixed at zero and cannot move, all the fields are also abbeys the symmetry conditions and number of the variables reduces considerably (in a spherical symmetry only the distance from the centre enters the equations). The standard minimization procedure of the free energy functional leads to a series of equations which are coupled with the fields which are the averages of the variables. In the following the example of three component system is described.
In case of solutions of two surfactants, or polymers and phospholipids, the free energy can be written in the following form:  where m index refers to phospholipids layer (membrane) and p index refers to polymer. The solvent in this theory is a continuous media that fills in the space between the species. The last term is the entropy associated with the solvent and is the concentration of the solvent. Note that the variables in bold representing the degrees of freedom of the chains inside the aggregate is split into (a,r), where a is the conformational states of a single chain (conformations) and r is the position of the conformations from the centre of the aggregate. The solvent molecule is assumed not to have internal degrees of freedom and thus the concentration of solvent is a function of a only. If the interactions between the species and distribution function of conformations of the chains are known, the free energy is settled and we can calculate all the equilibrium properties. In case of ideal chains, one could use the propagator of the Gaussian chain to fix the distribution of conformations . However there is no analytical expression for the chains with interactions and excluded volume. The SCMF Theory solves this problem by calculating exactly the conformations of single chains, both polymers and phospholipids. They are calculated using different Monte Carlo algorithms and the conformations are stored in a file. The equilibrium properties are calculated by averaging with the probability of the distribution in the file. Using standard for density functional theory minimization of the free energy with respect to Pm(a) and Pp(a) subject to incompressibility condition, that the sum of the polymer, phospholipids and solvent concentration should not exceed 1, we obtain the equilibrium probabilities of the conformations:
 Once the probabilities of conformations are calculated, the thermodynamic properties are known and are calculated as the averages over conformations, for example the average concentrations of the species is given by  where øi(a,r) is the concentrations of a given conformation at a given position. They are calculated only once from the configurational file. After all types of averages are calculated, they are substituted back into the expression to improve the probabilities and so forth according to self-consistency closure procedure. | 
