^{1,a)}, Günther H. Peters

^{2,b)}, Flemming Y. Hansen

^{3,c)}and Ole Hassager

^{1,d)}

### Abstract

We present a new framework for the description of macromolecules subject to confining geometries. The two main ingredients are a new computational method and the definition of a new molecular size parameter. The computational method, hereafter referred to the confinement analysis from bulk structures (CABS), allows the computation of equilibrium partition coefficients as a function of confinement size solely based on a single sampling of the configuration space of a macromolecule in bulk. Superior in computational speed to previous computational methods, CABS is capable of handling slits, channels, and box confining geometries for all molecular architectures. The new molecular size parameter, hereafter referred to the *steric exclusion radius* , is explicitly defined and computed for a number of rigid objects and flexible polymers. We suggest that is the relevant molecular size parameter for characterization of spatial confinement effects on macromolecules. Results for the equilibrium partition coefficient in the weak confinement regime depend only on the ratio of to the confinement size regardless of molecular details.

Y.W. acknowledges support by the Danish Research Council for Technology and Production Sciences Grant No. 26-04-0074. G.H.P. acknowledges financial support from the Danish National Research Foundation via a grant to MEMPHYS-Center for Biomembrane Physics. Simulations were performed at the Danish Center for Scientific Computing at the Technical University of Denmark.

I. INTRODUCTION

II. THEORY

A. A grand canonical ensemble formulation

B. The CABS method

C. The steric exclusion radius

III. SIMULATION

IV. RESULTS AND DISCUSSIONS

A. Bulk structure properties

B. Comparison with prior studies

C. A universal partitioning behavior

V. CONCLUSIONS

### Key Topics

- Polymers
- 96.0
- Macromolecules
- 38.0
- Surface acoustic waves
- 19.0
- Macromolecular conformation
- 15.0
- Monte Carlo methods
- 15.0

## Figures

A polymer chain of given configuration is confined in a slit of width and orientation . This problem is essentially one dimensional (1D). By projecting the polymer configuration onto , we obtain as the maximum covered distance in direction. In order for this configuration not to be intersected by the slit walls, the first bead can only be located in the region in the direction. The probability that this configuration is not intersected by the slit walls is .

A polymer chain of given configuration is confined in a slit of width and orientation . This problem is essentially one dimensional (1D). By projecting the polymer configuration onto , we obtain as the maximum covered distance in direction. In order for this configuration not to be intersected by the slit walls, the first bead can only be located in the region in the direction. The probability that this configuration is not intersected by the slit walls is .

Example of a 3D structure for a flexible linear RW polymer and its coarse-grained model: An elastic dumbbell.

Example of a 3D structure for a flexible linear RW polymer and its coarse-grained model: An elastic dumbbell.

Characteristic ratios , , and obtained from linear RW and SAW simulations are plotted as a function of the total number of beads, . Solid lines are predictions based on Eqs. (48)–(50).

Characteristic ratios , , and obtained from linear RW and SAW simulations are plotted as a function of the total number of beads, . Solid lines are predictions based on Eqs. (48)–(50).

Distribution functions of are shown for a rigid rod, an elastic dumbbell, and a linear RW polymer of beads. Points are our simulations results, and solid lines are predictions based on Eqs. (32), (35), and (43).

Distribution functions of are shown for a rigid rod, an elastic dumbbell, and a linear RW polymer of beads. Points are our simulations results, and solid lines are predictions based on Eqs. (32), (35), and (43).

Equilibrium partition coefficients as a function of are shown for a long chain linear RW polymer with (from top to bottom) a slit of width , a long square channel of cross section, a long cylinder of diameter , a cubic box of size , and a sphere of diameter . Both our simulation results and analytical results given in Eqs. (41) and (51) are shown for slit, square channel, and cubic box confining geometries, and they nearly coincide with one another even for as small as . For cylinder and sphere confining geometries, only Casassas results are shown (Ref. 1).

Equilibrium partition coefficients as a function of are shown for a long chain linear RW polymer with (from top to bottom) a slit of width , a long square channel of cross section, a long cylinder of diameter , a cubic box of size , and a sphere of diameter . Both our simulation results and analytical results given in Eqs. (41) and (51) are shown for slit, square channel, and cubic box confining geometries, and they nearly coincide with one another even for as small as . For cylinder and sphere confining geometries, only Casassas results are shown (Ref. 1).

Equilibrium partition coefficients as a function of are shown for linear RW polymers of beads with a slit confining geometry of width . From top to bottom: , and . The analytical solution for a rigid dumbbell (Ref. 14) is also plotted, which coincides with the simulation results of . Casassa’s result (Ref. 1) given by Eq. (41) corresponds to .

Equilibrium partition coefficients as a function of are shown for linear RW polymers of beads with a slit confining geometry of width . From top to bottom: , and . The analytical solution for a rigid dumbbell (Ref. 14) is also plotted, which coincides with the simulation results of . Casassa’s result (Ref. 1) given by Eq. (41) corresponds to .

Equilibrium partition coefficients as a function of are shown for linear SAW polymers of beads with a slit confining geometry of width . From top to bottom: , simulation results of a linear RW chain of , and Casassa’s result (Ref. 1) given by Eq. (41) corresponding to a linear RW chain of .

Equilibrium partition coefficients as a function of are shown for linear SAW polymers of beads with a slit confining geometry of width . From top to bottom: , simulation results of a linear RW chain of , and Casassa’s result (Ref. 1) given by Eq. (41) corresponding to a linear RW chain of .

Equilibrium partition coefficients as a function of are shown for linear RW polymers of beads with a slit confining geometry of width . From top to bottom: , and . Casassa’s result (Ref. 1) given by Eq. (41) corresponds to , which nearly coincides with the simulation results for .

Equilibrium partition coefficients as a function of are shown for linear RW polymers of beads with a slit confining geometry of width . From top to bottom: , and . Casassa’s result (Ref. 1) given by Eq. (41) corresponds to , which nearly coincides with the simulation results for .

Equilibrium partition coefficients as a function of are shown for linear SAW polymers of beads with a slit confining geometry of width . From top to bottom: , simulation results of a linear RW chain of , and Casassa’s result (Ref. 1) given by Eq. (41) corresponding to a linear RW chain of .

Equilibrium partition coefficients are shown (a) as a function of , (b) as a function of , (c) as a function of , and (d) as a function of for (from top to bottom) a linear SAW polymer of , a linear RW polymer of , a symmetric three-arm RW star polymer of arm length with , and a symmetric four-arm RW star polymer of arm length with with a slit confining geometry of width . In subfigure (d), are also shown for an elastic dumbbell, a rigid rod, and a sphere.

Equilibrium partition coefficients are shown (a) as a function of , (b) as a function of , (c) as a function of , and (d) as a function of for (from top to bottom) a linear SAW polymer of , a linear RW polymer of , a symmetric three-arm RW star polymer of arm length with , and a symmetric four-arm RW star polymer of arm length with with a slit confining geometry of width . In subfigure (d), are also shown for an elastic dumbbell, a rigid rod, and a sphere.

Equilibrium partition coefficients are shown (a) as a function of , (b) as a function of , (c) as a function of , and (d) as a function of for (from top to bottom) a linear SAW polymer of , a linear RW polymer of , a symmetric three-arm RW star polymer of arm length , and a symmetric four-arm RW star polymer of arm length with a square channel confining geometry of cross section. In subfigure (d), for a sphere is also shown.

Equilibrium partition coefficients are shown (a) as a function of , (b) as a function of , (c) as a function of , and (d) as a function of for (from top to bottom) a linear SAW polymer of , a linear RW polymer of , a symmetric three-arm RW star polymer of arm length , and a symmetric four-arm RW star polymer of arm length with a square channel confining geometry of cross section. In subfigure (d), for a sphere is also shown.

Equilibrium partition coefficients are shown (a) as a function of , (b) as a function of , (c) as a function of , and (d) as a function of for (from top to bottom) a linear SAW polymer of , a linear RW polymer of , a symmetric three-arm RW star polymer of arm length , and a symmetric four-arm RW star polymer of arm length with a cubic box confining geometry of size . In subfigure (d), for a sphere is also shown.

Equilibrium partition coefficients are shown (a) as a function of , (b) as a function of , (c) as a function of , and (d) as a function of for (from top to bottom) a linear SAW polymer of , a linear RW polymer of , a symmetric three-arm RW star polymer of arm length , and a symmetric four-arm RW star polymer of arm length with a cubic box confining geometry of size . In subfigure (d), for a sphere is also shown.

## Tables

The steric exclusion radius and root-mean-square radius of gyration for some simple rigid objects: A uniform sphere (sphere) and a spherical shell (shell) both of radius ; a thin rod (rod) and a rigid dumbbell (dumbbell) both of length ; a uniform thin disk (disk) and a rigid ring (ring) both of radius .

The steric exclusion radius and root-mean-square radius of gyration for some simple rigid objects: A uniform sphere (sphere) and a spherical shell (shell) both of radius ; a thin rod (rod) and a rigid dumbbell (dumbbell) both of length ; a uniform thin disk (disk) and a rigid ring (ring) both of radius .

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