Bistability in a self-assembling system confined by elastic walls. Exact results in a one-dimensional lattice model
aa r X i v : . [ c ond - m a t . s o f t ] N ov Bistability in a self-assembling system confined by elastic walls.Exact results in a one-dimensional lattice model.
J. P¸ekalski, A. Ciach, and N. G. Almarza Institute of Physical Chemistry, Polish Academy of Sciences, 01-224 Warszawa, Poland Instituto de Qu´ımica F´ısica Rocasolano,CSIC, Serrano 119, E-28006 Madrid, Spain (Dated: May 8, 2019)
Abstract
The impact of confinement on self-assembly of particles interacting with short-range attractionand long-range repulsion (SALR) potential is studied for thermodynamic states correspondingto local ordering of clusters or layers in the bulk. Exact and asymptotic expressions for thelocal density and for the effective potential between the confining surfaces are obtained for a one-dimensional lattice model introduced in [J. P¸ekalski et. al.
J. Chem. Phys. , 144903 (2013)].The simple asymptotic formulas are shown to be in good quantitative agreement with exact resultsfor slits containing at least 5 layers. We observe that the incommensurability of the system sizeand the average distance between the clusters or layers in the bulk leads to structural deformationsthat are different for different values of the chemical potential µ . The change of the type of defectsis reflected in the dependence of density on µ that has a shape characteristic for phase transitions.Our results may help to avoid misinterpretation of the change of the type of defects as a phasetransition in simulations of inhomogeneous systems. Finally, we show that a system confined bysoft elastic walls may exhibit bistability such that two system sizes that differ approximately bythe average distance between the clusters or layers are almost equally probable. This may happenwhen the equilibrium separation between the soft boundaries of an empty slit corresponds to thelargest stress in the confined self-assembling system. . INTRODUCTION Self-assembly is a fundamental process in living matter, it is even thought to be the keyto understand the origin of life as it leads to organization of intracell compartments [1]. Self-assembly is responsible for spontaneous formation of lipids into a bilayer. Moreover, it takespart in intercell communication when proteins adsorbed on a cell membrane spontaneouslyaggregate and form clusters which can perform functions unavailable to a single proteinmolecule [2–4]. On the other hand, the self-assembly of nanoparticles is of high interest inindustry. In nanotechnology self-assembly is a base for techniques aimed at device minia-turization and material production with novel electronic, mechanical and optical properties[5, 6].Heterogeneity on a mesoscopic length scale is often an effect of competing tendenciesin the pair interaction potential. In the case of nanoparticles or globular proteins there isa competition between solvent-induced attraction, and repulsion that is typically (but notexclusively) of electrostatic origin [4, 7, 8]. The effective isotropic short-range attraction(SA) between nanoparticles, ions or organic molecules favours their aggregation, while thepresence of the long-range isotropic repulsion (LR) effects in the separation of the aggregates.The competition between these opposite tendencies results in thermodynamic stabilizationof spatially inhomogeneous patterns made of globular or elongated clusters, or layers (stripesin a case of a surface). The clusters or layers are periodically distributed and form regularpatterns in ordered phases [9–12]. In the disordered phase the particles also self-assembleinto clusters or layers for some range of temperature and concentration [13–15]. However,in the disordered phase these objects are ordered only locally, and this short-range order isreflected in the exponentially damped oscillatory behavior of the correlation function on themesoscopic length scale [9, 11, 16, 17].In intracell compartments, in pores of a porous material or on geometrically patternedsurfaces, the soft or rigid boundaries of the system can have an ordering or disorderingeffect on the confined clusters or layers. The key factor is the commensurability of thetypical distance between the objects in the bulk, and the size of the compartment. Despitethe fact that the confinement plays a very important role in biological systems, in pores ofporous materials, and on patterned surfaces, the effects of confinement on the self-assemblingsystems have been much less studied than the bulk properties. In the case of the SALR2nteraction potential the impact of a slit-type confinement on thermodynamically stablepatterns on a surface (2d system) was studied by Monte Carlo simulations [18] and bydensity functional theory [9]. In Ref. [18] the authors found that unlike in the bulk system,in presence of neutral walls a switch from the cluster to the lamellar morphology withincreasing temperature is possible. Moreover, the orientation of the lamella depends on thedistance between the walls and the particle-wall interaction parameters. In Ref.[9] the authorfocused on determining the sequences of stable structures for increasing distance betweenthe walls at a given temperature and for fixed density. He confirmed that the change of thedistance can lead to the change of the stable-phase morphology, especially if the period ofthe structure stable in the bulk and the width of the slit are incommensurate.In this work we focus on these effects of confinement on the SALR systems that have notbeen investigated yet, although in our opinion play a very important role. We limit ourselvesto the disordered phase, and consider only permeable confining walls, i.e. the system caninterchange particles with a reservoir (Grand canonical ensemble). Our first question ishow the structural defects in the case of incommensurability between the system size andthe period in the bulk phase depend on thermodynamic state and on the interaction withthe surfaces. The second question concerns the fluid-induced effective interactions betweenthe confining surfaces for different values of the chemical potential. The periodic order onthe mesoscopic length scale can induce a periodic effective interaction between confiningsurfaces on the same length scale. This is analogous to the periodic solvation force onthe atomic length scale in simple fluids [19, 20]. In contrast to the amphiphilic systems,where the effective interaction was intensively investigated both experimentally [21, 22] andtheoretically [23–28], in the case of the SALR potential it has not been studied yet.In biological systems or in pores of a soft porous material the compartments are sur-rounded by lipid bilayers or by elastic material. The separation between the confiningsurfaces can be varied, and this change is associated with elastic energy. Mechanical equi-librium between the solvation force resulting from the stress in the confined self-assemblingsystem, and the elastic force resulting from deformation of the confining elastic material (e.g.lipid bilayer) determines the equilibrium shape of the system boundary. Changes of a ther-modynamic state, leading to the change of the solvation force, might lead to shape and/orsize transformations. In this work we address a question of equilibrium wall separation ina pore of an elastic material containing particles interacting with the SALR potential. It3ould be interesting to know the effect of the presence of particles on the equilibrium sepa-ration between the elastic walls, specially when the equilibrium thickness of the pore in theabsence of particules and the period of the bulk phase are incomensurate.In order to address the aforementioned questions on a general level we consider a genericlattice model of a SALR system that can be solved exactly in one dimension (1d). Our resultscan give some insight for the properties of two and three dimensional systems confined inslits. Moreover, a pseudo-1d system confined by elastic boundaries is formed, for example,by a long protrusion in a vesicle filled with charged nanoparticles. In Ref. [29] it was shownthat such a protrusion responds elastically to an external stress in direction parallel to itsaxis. The bulk properties of the model were thoroughly studied in Ref. [16]. In 1d theordered phases appear only at T = 0. The short-range order and the pretransitional effects,however, can be studied based on exact solutions. We found that in the disordered phasethe repulsion between the clusters or layers leads to a dependence of the average density ρ on the chemical potential µ or pressure p that is significantly different to that of simplefluids [16]. A characteristic plateau in ρ ( µ ) and ρ ( p ) appears when the density is equalto the density of the periodic structure that is favoured energetically. This plateau signalsthat a significant increase of pressure is necessary to overcome the repulsion between theclusters and to compress the system to a dense structure. In addition, for the range of µ corresponding to the plateau in ρ ( µ ) the correlation length is 3 or 4 orders of magnitudelarger than the size of the particles. The shape of the ρ ( p ) curve and the large correlationlength suggest that the solvation force can be quite strong in large pores even in the case ofthe short-range periodic order. We shall verify this expectation by exact results.In sec.2 we briefly describe the model introduced in Ref. [16] and the transfer matrixmethod. The details of the derivations are described in Appendices. In sec.3 we presentasymptotic expressions for the local density and for the effective potential between theconfining surfaces. We determine the range of validity of these formulas by comparison withexact results. In sec. 4 we discuss the dependence of the distribution of the particles insidethe pore on the chemical potential when the width of the slit and the period of the bulkstructure are incommensurate. We also compare the shape of ρ ( µ ) for various slits withthe result obtained in Ref.[16] for the bulk. In addition, we consider periodic boundaryconditions in the case of incommensurability, in order to help to interpret simulation results.The effective interaction for different thermodynamic states and the equilibrium width of a4ystem with elastic boundaries are determined in sec.5. The summary and conclusions arepresented in sec.6. II. THE MODEL AND THE METHOD OF EXACT SOLUTIONS
We consider a lattice model for systems with particles interacting with a short-rangeattraction and long-range repulsion (SALR) potential. We assume that the particles occupylattice sites on a 1 dimensional lattice, and the lattice constant a is comparable with theparticle diameter σ . The particles attract or repel each other when they are the nearestor the third neighbors respectively. The model with the same interaction potential andwith periodic boundary conditions (PBC) was solved exactly in Ref.[16] for the system sizes L = 6 N , where N is integer. For such system sizes the energetically favourable structureof 3 occupied sites separated by 3 empty sites is possible, and the properties of the bulksystem can be reproduced. For L = 6 N the incommensurability between the system size L and the period of the ordered structure may influence the results. Here we focus on theimpact of this incommensurability in the case of the PBC and in the case of a slit type ofconfinement, i.e. with rigid boundary conditions (RBC). In the case of the PBC we areinterested in how the incommensurability influences the dependence of the density ρ on thechemical potential µ for different system sizes. Our exact results may help to interpret theresults of simulations that are performed for finite systems when the incommensurabilitymight be a serious issue. In the case of the RBC we assume that the confining walls caninteract with the particles located at the first and the last site of the system (see Fig. 1).The confining walls represent real physical confinement e.g. in a porous material or in athin film on a solid substrate.We assume that the lattice consists of L sites labeled from 1 to L . In order to describewhether the lattice site x is occupied or not we introduce an occupation operator ˆ ρ ( x ) whichis equal to 1 or 0 respectively. Hence, the configuration of the system (the microstate) isgiven by { ˆ ρ ( x ) } ≡ ( ˆ ρ (1) , ..., ˆ ρ ( L )). The probability of the microstate { ˆ ρ ( x ) } is P [ { ˆ ρ ( x ) } ] = e − βH [ { ˆ ρ ( x ) } ] Ξ , (1)where Ξ is the Grand Partition function, β = ( k B T ) − , k B is the Boltzmann constant and5 IG. 1: Scheme of the model for a system of size L = 15. The lattice constant a is equal to theparticle diameter σ . The particles attract or repel each other with the energy − J or J when theyare the nearest or the third neighbors respectively. If a particle occupies the first or the last siteof the lattice then it interacts with the confining wall with the energy h or h L respectively. H is the thermodynamic Hamiltonian which contains the energy and the chemical potentialterm H [ { ˆ ρ } ] = 12 L X x =1 L X x ′ =1 ˆ ρ ( x ) V ( x − x ′ ) ˆ ρ ( x ′ ) + h ˆ ρ (1) + h L ˆ ρ ( L ) − µ L X x =1 ˆ ρ ( x ) , (2)where the particle-particle interaction potential is V (∆ x ) = − J for | ∆ x | = 1 , + J for | ∆ x | = 3 , J as the energy unit and introduce dimensionless variables for any quantity X with dimension of energy as X ∗ = X/J , in particular T ∗ = k B T /J , J ∗ = J /J , h ∗ = h /J , (4) h ∗ L = h L /J , µ ∗ = µ/J . (5)We solve the model exactly by the transfer matrix method, as in Ref. [16]. Becausethe range of the particle-particle interactions is 3, the transfer matrix operates between themicrostates in boxes that are located next to each other, and each box consists of 3 sites.There are 8 microstates in each box, therefore the dimension of the transfer matrix is 8.The system can be divided into such boxes when L = 3 N . In general, the expression for theGrand Partition function for L = 3 N + j depends on both, N and the reminder of divisionof L by 3, j = L mod , ,
2. We describe the method in more detail, and give the exactexpression for Ξ in Appendix A.When the correlation length between the particles is comparable with the distance be-tween the confining walls, then the distribution of the particles is influenced by both walls.6his leads to the excess of the grand potential depending on the distance between the walls[30], Ω ex ≡ Ω − Ω bulk = γ + γ L + Ψ( L ) (6)where Ω = − k B T ln Ξ and Ω bulk = − k B T ln Ξ bulk are the grand potential in the slit and inthe bulk of the same size L respectively, γ and γ L are the wall-fluid surface tensions andΨ( L ) corresponds to the effective interaction between the confining walls [30]. The effectiveforce between the surfaces is −∇ Ψ( L ). The exact expressions for γ , γ L and Ψ( L ) are givenin Appendix C.The expression for the local average density at the site x = 3 n + l in the system of size L = 3 N + j with j = 0 , , n and l = 1 , ,
3, and in addition on N and j . The rather complex formulas are given in Appendix B. The exact expressions take muchsimpler asymptotic forms for N ≫ n ≃ N/
2. We present the asymptotic formulas for N ≫ L ), and compare them with the exact results in the nextsection. III. ASYMPTOTIC EXPRESSIONS FOR LARGE SLITS AND THE RANGE OFTHEIR VALIDITY
In the energetically favourable structure clusters composed of 3 particles are separatedby 3 empty sites. For this reason the properties of the system confined in the slit of largewidth L = 3 N + j depend on both, the number of the triples of sites, N ≈ L/
3, and thenumber of the additional sites, j = 0 , ,
2. Let us first consider the average local densityin the central part of the slit of large width, N ≫
1. We divide the system into triples ofsites. Each site x is characterized by the number of the triple to which it belongs, n , and theposition inside the triple, l , so that x = 3 n + l with l = 1 , ,
3. The expression for h ˆ ρ (3 n + l ) i depends on n and l , as well as on N and j . From the exact formulas given in Appendix Bwe obtain the asymptotic expression for N → ∞ and n ≃ N/ h ˆ ρ (3 n + l ) i ≃ ¯ ρ + A ( l ) cos( nλ + θ ( l )) e − n/ξ + A L ( l ) cos(( N − n ) λ + θ L ( l )) e − N − n ) /ξ . (7)The explicit expressions for ¯ ρ , the amplitudes A ( l ) , A L ( l ) and the phases θ ( l ) , θ L ( l ) aregiven in Appendix B (these quantities depend also on N and j ). The decay length ξ is given7y the same expression as the correlation length in the bulk [16], ξ = 3 / ln λ | λ | ! , (8)where λ and λ = | λ | exp( iλ ) are the largest and the second largest eigenvalues of thetransfer matrix. The transfer matrix is not Hermitian, and some of the eigenvalues can becomplex. The presence of the imaginary part of λ depends on J ∗ , and on the thermodynamicstate. The monotonic decay of the density near a single surface occurs when λ is real andpositive ( λ = 0). The exponentially damped periodic structure with the period 6 occurswhen λ is real and negative, ( λ = π ). In most cases, however, including J ∗ = 3 for therange of µ ∗ studied in this work, λ is complex and the period of the damped oscillations isnoninteger.The asymptotic formula for the effective interaction potential for N → ∞ is β Ψ(3 N + j ) ≃ A ( j ) cos( λN + φ ( j )) e − N/ξ . (9)The explicit expressions for the amplitude A ( j ) and the phase φ ( j ) are given in AppendixC. The asymptotic formulas are simply the exponentially damped periodic functions. Similarexpressions, but without the amplitude modulations, were obtained in mean-field theoriesof confined self-assembling systems [26, 31–33]. These rather simple asymptotic forms arestrictly valid for N ≫ n ≃ N/
2. We check the validity of the asymptotic expressionsby comparing them with the exact results. The exact and asymptotic formulas are valid forany J ∗ . To fix attention we focus in this paper on the case of strong repulsion, J ∗ = 3, inthe analysis of mechanical and structural properties of the confined self-assembling system.As shown in Fig.2, the agreement of the asymptotic expression for the local density withthe exact result is very good already for L = 42, and the discrepancy between the exactand asymptotic expressions appear only very close to the surface. For L = 30 the accuracyof the asymptotic expression is less good but it is still satisfactory, except from the clustersadsorbed at the surfaces, where some discrepancy can be observed. Thus, the asymptoticformula is sufficiently accurate not only in the center, but inside the whole slit for slitscontaining 5 or more clusters.In the asymptotic expressions the decay length and the period of oscillations of the localdensity in the slit and the correlation function in the bulk are the same. In Fig.3 we compare8 IG. 2: Comparison of the exact (18) and approximate (7) formulas for the average density for J ∗ = 3, µ ∗ = 0 , T ∗ = 0 . h ∗ = h ∗ L = −
1. Upper panel L = 30, lower panel L = 42. the exact results for the local density and for the correlation function. In order to comparethe two functions, we add the average density of the bulk system to the linearly scaledcorrelation function, and obtain good agreement for the distance from the surface z > L ) takes almostequal values for two consecutive system sizes. However, for the exact and the approximateformulas this phenomenon occurs for different system sizes eg. in Fig. 4 for L = 39 ,
40 forthe exact result, and for L = 27 ,
28 for the approximate formula.9
IG. 3: Comparison of the density profile in a slit (black dashed line) for J ∗ = 3, µ ∗ = − . T ∗ = 0 . h ∗ = h ∗ L = − L = 96, and the bulk correlation function obtained in [16] (redsolid line) for the same thermodynamic state. The correlation function was linearly scaled andshifted by the average density of the bulk system, ρ = 0 . L ) for J ∗ = 3, µ ∗ = 0, T ∗ = 0 . h ∗ = h ∗ L = − IV. EFFECTS OF INCOMMENSURABILITY OF THE SYSTEM SIZE AND THEPERIOD OF THE BULK STRUCTURE
In this section we study the effect of the incommensurability of the system size and theperiod of the bulk structure on the distribution of the particles and on the dependence of theaverage density on the chemical potential. Our aim is to verify how ρ ( µ ∗ ) is influenced bythe presence of structural defects that must be present in the case of the incommensurability.10e first consider the PBC, and next the RBC. A. The case of periodic boundary conditions (PBC)
We focus on L = 6 N + 3, i.e. on the largest mismatch between L and 6 (the low- T periodin the ordered phase). Let us first investigate the ground state (GS), i.e. the case of T ∗ = 0. FIG. 5: Scheme of the ground state ( T ∗ = 0) for J ∗ = 3 for the bulk system ( L = 6 N ) (a) and forthe system of size L = 9 (b). For µ ∗ < − / µ ∗ > / − / < µ ∗ < / L = 9the stability region of the periodic phase is split into µ ∗ < µ ∗ > For L = 6 N + 3 we may expect that in the periodic phase either a separation between someclusters is larger than 3, or some clusters are larger than 3 (see Fig.5). When the separationbetween the clusters is larger than 3 and we add one particle to a cluster consisting of at least3 particles, then the increase of the Hamiltonian is ∆ H ∗ = − J ∗ − µ ∗ . For µ ∗ < J ∗ − µ ∗ > J ∗ − H ∗ > H ∗ < H ∗ the voids in the first case and the clusters in the secondcase occupy 3 more sites. At T ∗ = 0 the average density jumps by 3 /L for µ ∗ = J ∗ −
1. TheGS in the bulk ( L = 6 N ) and for L = 9 is shown in Fig.5 for J ∗ = 3. Note that we have∆ H ∗ = − J ∗ − µ ∗ = 0 for µ ∗ = 2 in this case, therefore for µ ∗ = 2 the GS is degenerate,and the cluster can consist of either 3,4,5 or 6 particles.In Fig.6 we show ρ ( µ ∗ ) for T ∗ = 0 .
3. Note that the transition between the two types ofdefects, i.e. larger voids or larger clusters could be misinterpreted as a transition between dif-ferent phases, because when a system undergoes a first-order phase transition, steps in ρ ( µ ∗ )appear. Thus, the results of simulations in the case of systems with spatial inhomogeneities11n a mesoscopic length scale should be interpreted with special care, especially when severalperiodic phases with different periods can appear. However, in the case of structural defectsthe height of the step decreases as ∼ /L for increasing L , and for certain system sizes thestep disappears (Fig.6). This observation may help to interpret the simulation results. FIG. 6: The average density ρ ∗ as a function of the chemical potential µ ∗ for J ∗ = 3 and T ∗ = 0 . L = 9 (red dashed line), L = 10 . . .
14 (thin color lines) and L = 15(black solid line). B. The case of rigid boundary conditions (RBC)
We first focus on attractive surfaces. Let us consider the Hamiltonian for a single clustercomposed of n ≤ H ∗ = h ∗ − ( n − − nµ ∗ . The adsorptionof the cluster is energetically favourable compared to vacuum for µ ∗ > ( h ∗ + 1 − n ) /n . Inorder to fix attention, we assume that the interaction with the surfaces is the same as theparticle-particle attraction, h ∗ = h ∗ L = −
1. In this case a cluster adsorbed at each attractivesurface is energetically favourable for µ ∗ > −
1. Thus, for µ ∗ > − L = 6 N when both surfaces are attractive. For L = 6 N + 3 the GS of the system is degenerate in thewhole stability region of the periodic phase, because the defects in the periodic structure thatare caused by the incommensurability of the period and the system size are not localized.Moreover, the stability region of the periodic phase splits into 4 regions, corresponding todifferent numbers and sizes of the clusters present in the slit. We choose L = 19 and presenttypical microscopic states of the GS in Fig.7. For µ ∗ < < µ ∗ < J ∗ − J ∗ − < µ ∗ < J ∗ −
1) there are 4 clusters consisting of 3particles. Finally, for µ ∗ > J ∗ −
1) there are 3 clusters separated by two voids composedof 3 empty sites, and each cluster consists of at least 3 particles.
FIG. 7: Typical microstates in the degenerate GS for a slit of size L = 19 with attractive walls.The range of the chemical potential corresponding to the shown microstates is (a) − / < µ ∗ < < µ ∗ < J ∗ −
1, (c) J ∗ − < µ ∗ < J ∗ −
1) and (d) 2( J ∗ − < µ ∗ < J ∗ − / In Fig.8 we present ρ ( µ ∗ ) for the slit with L = 19 at T ∗ = 0 .
1. The average densitiescorresponding to the plateaus are shown in the insets. Note the similarity between the
FIG. 8: Density ρ as a function of the dimensionless chemical potential µ ∗ for J ∗ = 3 and T ∗ = 0 . L = 19 with attractive walls. For increasing µ ∗ we first observe the adsorption,and next 4 plateaus. The plateaus from (a) to (d) correspond to the average densities shown inthe insets. The steps between them occur for µ ∗ ≈ , ,
4, i.e. near the GS coexistence betweendifferent structures in confinement (see Fig.7). ρ ( µ ∗ ) represent a physical effect, namelystructural changes such as a jump of a number of the clusters or a change of their size as afunction of the chemical potential. Such abrupt changes in a slit induced by small changesin the surroundings occur when the size of the system and the period of the bulk phase areincommensurate.Let us focus on the role of the interaction with the confining surfaces. The attractive andrepulsive surfaces are compared in Fig. 9 for a large slit. As expected, when the walls arerepulsive we do not observe the step in ρ ( µ ∗ ) at µ ∗ ≈ − − . < µ ∗ < − . µ ∗ > − .
45 - there is one more cluster,and one more step in ρ ( µ ∗ ) in the slit with the attractive surfaces for L = 20 , , , . . . . V. EFFECTIVE INTERACTION BETWEEN THE WALLS AND DEFORMA-TIONS OF ELASTIC CONTAINERS
In this section we discuss the effective potential between the confining surfaces separatedby the distance L . We first consider walls separated by a fixed distance. Next we assume thatthe walls are elastic, and the change of the wall separation is possible at the cost of elasticenergy. When the equilibrium width of the empty slit, L , and the period of the orderedphase do not match, the elastic energy and the fluid-induced stress are in competition. Weask how the equilibrium width of the slit filled with the inhomogeneous fluid differs from L . A. The case of fixed distance between the confining walls
The exact results for the effective potential between the confining walls separated by afixed distance, Ψ( L ), are presented in Fig. 10 for the chemical potential corresponding to theGS stability of the vacuum, the periodic phase and the dense phase (compare Fig.5). Note14 IG. 9: Panel (a) ρ ∗ ( µ ∗ ) for J ∗ = 3, T ∗ = 0 .
03 and L = 50 for systems with PBC (dashed line),RBC with attractive walls (dotted line) and RBC with repulsive walls (solid line). The rapidchange of density at µ ∗ ≈ − µ ∗ = − .
55 and µ ∗ = − . µ ∗ ≈ − .
45 when the walls are attractive. that the confined fluid leads to repulsion or attraction between the walls when the dilute orthe dense pseudo-phase is stable in the bulk respectively. The repulsion may follow from theadsorption of the clusters at the surfaces, since the clusters repel each other. The oscillationsof Ψ( L ) are present if the periodic distribution of clusters is thermodynamically preferred.These oscillations should be interpreted as follows: the minima of Ψ( L ) correspond to thesystem sizes commensurate with the periodic structure, therefore if we would allow thesystem to shrink or expand, then in order to suppress the internal stress the system would15hange its size to the value corresponding to the nearest minimum of Ψ( L ). The bigger is theslope of the oscillations, the stronger is the effective force leading to the nearest minimumof Ψ( L ).For large L the decay rate of Ψ( L ), ξ , is equal to the bulk correlation length (see Eq.(9)).In Ref. [16] it was shown that the correlation length in the considered model can be afew orders of magnitude larger than the molecular size for µ ∗ corresponding to the stabilityregion of the periodic phase on the GS ( − / < µ ∗ < / J ∗ = 3). In Fig. 11 we showthat for µ ∗ = 2, where ξ takes the maximum, Ψ( L ) ∼ . k B T even for system sizes 4 ordersof magnitude larger than the particle diameter. FIG. 10: Ψ( L ) for J ∗ = 3 and T ∗ = 0 . µ ∗ and bothwalls attractive. A) µ ∗ = − µ ∗ = 0, C) µ ∗ = 4, D) µ ∗ = 5. L is in units of the particlediameter σ FIG. 11: Ψ( L ) for J ∗ = 3, T ∗ = 0 . µ ∗ = 2 for non-interacting walls. L is in units of theparticle diameter σ . The case of elastic confining walls We assume that the width L of the slit can oscillate around L = L , where L is theequilibrium width in the absence of particles inside the pore. This oscillation can be con-trolled be a harmonic potential energy U w ( L ) = k · ( L − L ) (see Fig. 12). Next we assumethat when the slit is in contact with the reservoir of particles, and the chemical potential µ ∗ and temperature T ∗ are fixed, then in mechanical equilibrium the sum of U w ( L ) and theparticle-induced effective potential Ψ( L ) takes the minimum. We should note that similarassumptions lead to correct prediction of swelling of microporous carbons induced by ad-sorption of argon [34]. Here we make a similar assumption for larger particles and systemsizes, and softer confining surfaces. In Fig. 13 we present the sum of Ψ and U w as a functionof the system size. Note that when L corresponds to the maximum of Ψ( L ), i.e. to a largestress induced by the confined fluid, then U w ( L )+Ψ( L ) may have two minima of very similardepth for wall separations that differ approximately by the period of the bulk structure. Thenumber of clusters in these two states differs by one. As can be seen in Fig.13, the barrierbetween the two minima is of order of k B T for the assumed elastic constant k = 0 . k B T /σ . FIG. 12: Illustration of the system with elastic walls with spring constat k . In Fig.14 we show how the bistability appears when the chemical potential changes from µ ∗ = 0 or µ ∗ = 4 towards µ ∗ = 2. The barrier between the two minima decreases forincreasing | µ ∗ − | . Thus, by changing the concentration of particles in the surroundingswe can change the hight of the barrier and induce or suppress the jumps between the twowidths of the confined system. VI. SUMMARY
We have solved exactly the 1d model of a system interacting with the SALR potentialin slits of various widths. The distribution of the particles in confinement and the effec-tive potential between the confining surfaces have been calculated for different values of thechemical potential, from dilute to dense systems in the bulk. We paid particular attention17
IG. 13: The sum of the elastic energy of the confining boundaries and the effective interactioninduced by the confined self-assembling system for different equilibrium width of the empty slit L . J ∗ = 3, T ∗ = 0 . µ ∗ = 2, h ∗ = h ∗ L = −
1, and the spring constant k = 0 . k B T /σ , where σ isthe particle diameter.FIG. 14: The sum of the elastic energy of the confining boundaries and the effective interactioninduced by the confined self-assembling system for various values of µ ∗ . J ∗ = 3, T ∗ = 0 . h ∗ = h ∗ L = − L = 24, and the spring constant k = 0 . k B T /σ . to µ ∗ corresponding to inhomogeneous distribution of the particles in the bulk. We also ob-tained ρ ( µ ∗ ) for various system sizes and different short-range interactions with the surfaces.We paid particular attention to the system sizes incommensurate with the typical distancebetween the clusters or layers in the bulk.The most interesting result is the bistability of the system confined by elastic walls18Fig.13). The bistability occurs when the width in the absence of particles correspondsto the largest stress in the confined self-assembling system. The system choses n or n + 1layers with almost equal probability. The size difference between the two cases is similar tothe period of the bulk structure that in the case of colloids can be as large as hundreadsof nanometers or even micrometers. Similar phenomenon occurs in very narrow slits whenthe width is such that n and n + 1 atomic layers of the adsorbed gas are equally probable.The size difference, however, is of order of an angstrom. An interesting property is thepossibility of inducing or suppressing the bistability by changing the chemical potential, i.e.the concentration of particles in the surroundings.The confined self-assembling system behaves as a soft elastic material itself (Fig.10), andthe bistability takes place when its elastic constant is similar to the elastic constant of theboundaries. Such soft boundaries are formed in particular by biological membranes.Another interesting result is the dependence of the deformations in the confined systemon the conditions in the surroundings. We found that by changing µ ∗ we induce changesin the number and size of the layers in the confined system. These structural changes arereflected in “steps” in ρ ( µ ). In order to help to interpret simulation results we obtained exactexpressions for ρ ( µ ∗ ) in the case of PBC and various system sizes. We obtained steps in ρ ( µ ∗ )corresponding to the change of the type of defects resulting from the incommensurability.In order to avoid misinterpretation of these steps as phase transitions, one should verify ifthe steps disappear for some (commensurate) system sizes, and if their heights decays as ∼ /L for increasing system size L .Recently close similarity between the bulk properties of the SALR and the amphiphilicsystems has been demonstrated in Ref. [35, 36]. Based on this similarity we may expect thatour results concern also amphiphilic systems in confinement, but this expectation should beverified.Finally, we should note that our exact results concern open systems in contact with aparticle reservoir. Recently hard discs confined by a ring of particles trapped in holographicoptical tweezers, which form a flexible elastic wall were investigated [37]. For a fixed numberof confined particles a bistable state of a hexagonal structure and concentrically layered fluidmimicking the shape of the confinement was found. This phenomenon has some similarity toour bistability, since in both cases the adaptive confinement plays a crucial role. However, thefixed number of confined particles may alter the properties of the system confined between19daptive boundaries. Some of the lipid bilayers in living cells are permeable for proteins,while some other ones are not. In a forthcoming paper we shall compare the open and closedconfined systems with the same average number of particles.20 II. APPENDIXA. Partition function
Since the range of particle-particle interactions is 3, we introduce boxes consisting ofthree neighboring lattice sites. For the system of size L = 3 N + j , where j = 0 , ,
2, theboxes can be labeled by integer k = 1 , , . . . N . The microstates in the k -th box areˆ S ( k ) = ( ˆ ρ (3 k − , ˆ ρ (3 k − , ˆ ρ (3 k )) . (10)For N > L ≥
6) the Hamiltonian can be written in the form H ∗ [ { ˆ ρ } ] = ˆ ρ (1) h ∗ + ˆ ρ ( L ) h ∗ L + H ∗ j [ ˆ S ( N )] + N − X k =1 H ∗ t [ ˆ S ( k ) , ˆ S ( k + 1)] . (11)where H ∗ t [ ˆ S ( k ) , ˆ S ( k + 1)] = k X x =3 k − (cid:2) − ˆ ρ ( x ) ˆ ρ ( x + 1) + J ∗ ˆ ρ ( x ) ˆ ρ ( x + 3) − µ ∗ ˆ ρ ( x ) (cid:3) . (12)contains the interaction between two neighboring boxes and the chemical potential term inthe first box, H ∗ j [ ˆ S ( N )] = − ( P i =0 ˆ ρ (3 N − i ) ˆ ρ (3 N − i − − µ ∗ ( P i =0 ˆ ρ (3 N − i )) if j = 0 − ( P i =0 ˆ ρ (3 N +1 − i ) ˆ ρ (3 N − i )) + J ∗ ˆ ρ (3 N −
2) ˆ ρ (3 N +1) if j = 1 − µ ∗ ( P i =0 ˆ ρ (3 N + 1 − i )) − ( P i =0 ˆ ρ (3 N +2 − i ) ˆ ρ (3 N + 1 − i ))+ if j = 2 J ∗ ( P i =0 ˆ ρ (3 N − i ) ˆ ρ (3 N +1 + i )) − µ ∗ ( P i =0 ˆ ρ (3 N + 1 − i ))contains the particle-particle interactions between the particles which occupy the sites withinthe N -th box, and in addition the interactions between the particles at the sites labeled 3 N +1and 3 N +2 (if such sites exist for given L ). Finally, ρ (1) h ∗ and ρ ( L ) h ∗ L are the energies ofinteraction between the particles and the two walls. For N = 1 the Hamiltonian does notcontain the last term in (11). We consider only N > × T with the matrixelements T ( ˆ S ( k ) , ˆ S ( k + 1)) ≡ e − β ∗ H ∗ t [ ˆ S ( k ) , ˆ S ( k +1)] . (13)21he partition function in terms of the transfer matrix has the following formΞ = X ˆ S (1) ′ X ˆ S ( N ) e β ∗ ˆ ρ (1) h ∗ T N − [ ˆ S (1) , ˆ S ( N )] e β ∗ ˆ ρ ( L ) h ∗ L e β ∗ H ∗ j [ ˆ S ( N )] , (14)where P ′ ˆ S ( N ) denotes ′ X ˆ S ( N ) = P ˆ S ( N ) if j = 0 P ˆ S ( N ) P ˆ ρ (3 N +1) if j = 1 P ˆ S ( N ) P ˆ ρ (3 N +1) P ˆ ρ (3 N +2) if j = 2We transfer T to the base in which it is diagonal and the matrix elements of T N − can beeasily expressed by the sum over the eigenvalues λ k and the matrix elements P k ( ˆ S ( n )) ofthe matrix transforming T to its eigenbasis T N − ( ˆ S ( n ) , ˆ S ( m )) = X k =1 P k ( ˆ S ( n )) λ N − k P − k ( ˆ S ( m )) . (15)Hence the partition function isΞ = X ˆ S (1) ′ X ˆ S ( N ) 8 X k =1 e β ∗ ˆ ρ (1) h ∗ P k ( ˆ S (1)) λ N − k P − k ( ˆ S ( N )) e β ∗ ˆ ρ ( L ) h ∗ L e β ∗ H ∗ j [ ˆ S ( N )] . (16) B. Average density at a given site
The framework of the transfer matrix allows us to find a formula for average density atthe site x = 3 n + l , where n is the number of the triple to which the x -th site belongs and l = 1 , , < n < N the average density at the x -th site is h ˆ ρ ( x ) i = 1Ξ X ˆ S ( n ) X ˆ S (1) ′ X ˆ S ( N ) e β ∗ ˆ ρ (1) h ∗ T n ( ˆ S (1) , ˆ S ( n )) ˆ ρ ( x ) T N − ( n +1) ( ˆ S ( n ) , ˆ S ( N )) e β ∗ ˆ ρ ( L ) h ∗ L e β ∗ H ∗ j [ ˆ S ( N )] . (17)In terms of the eigenvalues it takes the form h ˆ ρ ( x ) i = 1Ξ X ˆ S ( n ) ˆ ρ ( x ) X ˆ S (1) e β ∗ ˆ ρ (1) h ∗ X k =1 P k ( ˆ S (1)) λ nk P − k ( ˆ S ( n )) · ′ X ˆ S ( N ) e β ∗ ˆ ρ ( L ) h ∗ L e β ∗ H ∗ j ( ˆ S ( N )) 8 X k =1 P k ( ˆ S ( n )) λ N − ( n +1) k P − k ( ˆ S ( N )) . (18)22e introduce the notation: a k + ib k ≡ X ˆ S (1) e β ∗ ˆ ρ (1) h ∗ P k ( ˆ S (1)) P − k ( ˆ S ( n )) , (19) c k + id k ≡ ′ X ˆ S ( N ) e β ∗ ˆ ρ ( L ) h ∗ L e βH ∗ j ( ˆ S ( N )) P k ( ˆ S ( n )) P − k ( ˆ S ( N )) (20)where λ ∈ R is the eigenvalue with the largest absolute value and i = √−
1. The dependenceof a k , b k , c k and d k on ˆ S ( n ) is not indicated for clarity of notation. The parameters c k and d k depend also on j = L mod 3. Then eq. (18) takes form h ˆ ρ (3 n + l ) i = λ N − Ξ X ˆ S ( n ) ˆ ρ (3 n + l ) X k =1 (cid:18) λ k λ (cid:19) n ( a k + ib k ) ! X k =1 (cid:18) λ k λ (cid:19) N − n − ( c k + id k ) ! . (21)Our aim is to obtain an asymptotic expression for h ˆ ρ ( x ) i for N → ∞ and n ∼ N/
2. Wesort the eigenvalues in the descending order of their absolute values and neglect in eq.(18)all the eigenvalues except from the first 3 of them. We limit ourselves to the two cases: 1) λ = ¯ λ = | λ | e iλ and 2) λ , λ ∈ R with | λ /λ | n ≪ n ≫ λ = ¯ λ then after some algebra we obtain h ˆ ρ (3 n + l ) i ≃ λ N − Ξ X ˆ S ( n ) ˆ ρ (3 n + l ) (cid:16) a c + 2 c (cid:18) | λ | λ (cid:19) n ( a cos( nλ ) − b sin( nλ )) + (22)2 a (cid:18) | λ | λ (cid:19) N − n − ( c cos(( N − n − λ ) − d sin(( N − n − λ )) (cid:17) In deriving (22) we took into account that (cid:16) | λ | λ (cid:17) n · (cid:16) | λ | λ (cid:17) ( N − n − ≪ (cid:16) | λ | λ (cid:17) N − n − for N ≫ n ∼ N/
2. Eq.(22) can be written in the form (7) with ξ defined in Eq.(8), λ definedbelow Eq.(8), and with the following expressions for the remaining parameters:¯ ρ ≡ λ N − Ξ X ˆ S ( n ) ˆ ρ (3 n + l ) a c , (23) A ( l ) = w if λ , λ ∈ R and | λ /λ | n ≪ w cos θ ( l ) if λ = ¯ λ A L ( l ) = w if λ , λ ∈ R and | λ /λ | n ≪ w exp (3 /ξ )cos θ L ( l ) if λ = ¯ λ θ ( l ) ≡ arctan w w , θ L ( l ) ≡ arctan w w − λ w ≡ λ N − Ξ X ˆ S ( n ) ˆ ρ (3 n + l ) a c , w ≡ λ N − Ξ X ˆ S ( n ) ˆ ρ (3 n + l ) b c , (24) w ≡ λ N − Ξ X ˆ S ( n ) ˆ ρ (3 n + l ) a c , w ≡ λ N − Ξ X ˆ S ( n ) ˆ ρ (3 n + l ) a d , (25)The above asymptotic expressions are not valid when λ and λ are both real, and | λ /λ | n = O (1). For the range of parameters studied in this article, however, λ and λ are complex conjugate numbers. C. Surface tension and effective interaction between the confining walls
The grand thermodynamic potential for the bulk system of the size L = 3 N + j , where j = 0 , ,
2, and N → ∞ is β Ω bulk ≃ N →∞ − L N ln λ N = − ln λ N − j λ . (26)The impact of the system geometry and the particle-wall interactions can be expressed byan excess grand potential Ω ex ≡ Ω − Ω bulk . We obtain the grand potential Ω of the confinedsystem using eq. (16): β Ω = − ln Ξ = − ln (cid:16) X k =1 λ N − k C k ( j ) (cid:17) , (27)where C k ( j ) = X S (1) ′ X ˆ S ( N ) e β ∗ ˆ ρ (1) h ∗ P k ( ˆ S (1)) P − k ( ˆ S ( N )) e β ∗ ˆ ρ ( L ) h ∗ L e β ∗ H ∗ j ( ˆ S ( N )) . (28)From (26)-(28) we obtain β Ω ex ≃ j λ − ln C ( j ) − ln (cid:16) X k =2 C k ( j ) C ( j ) (cid:16) λ k λ (cid:17) N − (cid:17) (29)The sum of the surface tensions and the effective potential between the confining surfacesin Eq.(6) are given by β ( γ + γ ) = ln λ − ln C (0) (30)and β Ψ( L ) = − ln (cid:16) X k =2 C k ( j ) C ( j ) (cid:16) λ k λ (cid:17) N − (cid:17) (31)24espectively, since we have verified that the sum of the first two terms in Eq. (29) does notdepend on j . In the asymptotic region of N → ∞ the above expression for Ψ( L ) takes theasymptotic form given in Eq.(9) with φ ( j ) = φ ( j ) − λ and with A ( j ) = C a ( j ) e /ξ if λ , λ ∈ R and | λ /λ | N ≪ C a ( j ) e /ξ if λ = ¯ λ , where C ak ( j ) e iφ k ( j ) = − C k ( j ) C ( j ) . Acknowledgment
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