Abstract
We determine the equilibrium chain end probability density \(P_N(x)\) for a one-dimensional ideal chain of N monomers grafted to a planar surface. This distribution is also the distribution function of a three-dimensional chain where the tangential dimensions y, z are integrated out of the distribution. With a small modification of the analysis of Erukhimovich, Johner, and Joanny for free chains near a wall, we are able to obtain exact results for \(P_N(x)\) for any monomer number N though with the restriction that the monomer–monomer bonds lengths are exponentially distributed. In particular, we obtain exact values for \(P_N\) and its derivative \(\textrm{d}P_N/\textrm{d}x\) at the surface for any N and the full profile \(P_N(x)\) for selected values of N. To determine the effect of the bond distribution, we find \(P_N(x)\) numerically for Gaussian and uniformly distributed one-dimensional bonds and compare with the exact results for exponential bonds. We suggest several ways to quantify the effect of bond type based both on \(P_N (x)\) near the surface and in the scaling region \(x\sim \sqrt{N} a\). We then extract the large N limit of \(P_N(x)\) and show that it is similar to the chain end probability for continuous chains but shifted toward smaller x. We show the amount of shift is a measure of the magnitude of the correction to the continuous chain \(P_N(x)\). We derive the value of the shift for exponential bonds and show that the value is different for other bond types. We argue the shift can be interpreted as an effective surface behind the actual surface.
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References
P. Pincus, Colloid stabilization with grafted polyelectrolytes. Macromolecules 24(10), 2912–2919 (1991)
J. Klein, Shear, friction, and lubrication forces between polymer-bearing surfaces. Annu. Rev. Mater. Sci. 26, 581–612 (1996)
T. Kreer, Polymer-brush lubrication: a review of recent theoretical advances. Soft Matter 12(15), 3479–3501 (2016). https://doi.org/10.1039/c5sm02919h
S. Alexander, Adsorption of chain molecules with a polar head: a scaling description. J. Phys. France 38(8), 983–987 (1977). https://doi.org/10.1051/jphys:01977003808098300
P.G. de Gennes, Configurations of polymers attached to an interface. Macromolecules 27, 435 (1980)
S.T. Milner, T.A. Witten, M.E. Cates, Theory of the grafted polymer brush. Macromolecules 21(8), 2610–2619 (1988). https://doi.org/10.1021/ma00186a051
T. Kreer, S. Metzger, M. Müller, K. Binder, Static properties of end-tethered polymers in good solution: a comparison between different models. J. Chem. Phys. 120(8), 4012–4023 (2004)
K. Binder, A. Milchev, Polymer brushes on flat and curved surfaces: How computer simulations can help to test theories and to interpret experiments. J. Polym. Sci. Part B Polym. Phys. 50(22), 1515–1555 (2012). https://doi.org/10.1002/polb.23168
E.A. DiMarzio, F.L. McCrackin, One-dimensional model of polymer absorption. J. Chem. Phys. 43(2), 539–547 (1965)
M. Adamuti-Trache, W.E. McMullen, J.F. Douglas, Segmental concentration profiles of end-tethered polymers with excluded-volume and surface interactions. J. Chem. Phys. 105(11), 4798–4811 (1996). https://doi.org/10.1063/1.472991
S. Eisenhaber, G. Zifferer, Comparative analysis of tethered and untethered polymers close to a surface investigated by Monte Carlo techniques. Macromol. Theory Simul. 24(6), 543–555 (2015). https://doi.org/10.1002/mats.201500032
Shi Ho Kim and Tae-Hee Lee, Conformation dynamics of poly(t) single stranded DNA at the single-molecule level. J. Phys. Chem. Lett. 12, 4576–4584 (2021)
M. Doi, S.F. Edwards, Theory of Polymer Dynamics (Clarendon Press, Oxford, 1988)
I.Y. Erukhomovich, A. Johner, J.F. Joanny, The ideal polymer chain near planar wall beyond the Dirichlet boundary condition. Eur. Phys. J. E 27, 435 (2008). https://doi.org/10.1140/epje/i2008-10392-5
I.Y. Erukhomovich, A. Johner, J.F. Joanny, Depletion of ideal polymer chains near a spherical colloid particle beyond the Dirichlet boundary conditions. Eur. Phys. J. E 31, 115 (2019). https://doi.org/10.1140/epje/i2010-10568-4
M.W. Matsen, J.U. Kim, A.E. Likhtman, Finite-n effects for ideal polymer chains near a flat impenetrable wall. Eur. Phys. J. E 29, 107 (2009). https://doi.org/10.1140/epje/i2009-10454-2
M.W. Matsen, P. Mahmoudi, Segregation of chain ends to the surface of a polymer melt. Eur. Phys. J. E (2014). https://doi.org/10.1140/epje/i2014-14078-1
W. Feller, An Introduction to Probability Theory and Its Applications, vol. I, 2nd edn. (Wiley, New York, 1957)
S. Chandrasekhar, Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89 (1943)
S. Redner, A Guide to First Passage Processes (Cambridge University Press, Cambridge, 2001)
P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979)
We used the Maple mathematics package for our calculations
G.B. Arfken, H.J. Weber, Essential Mathematical Methods for Physicists (Academic Press, San Diego, 2003)
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CY and BF initiated the research. CY did the calculations. BF provided discussions and helped with the manuscript.
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Yeung, C., Friedman, B.A. Exact results and the effect of monomer–monomer bond type for a grafted ideal chain. Eur. Phys. J. E 45, 84 (2022). https://doi.org/10.1140/epje/s10189-022-00239-6
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DOI: https://doi.org/10.1140/epje/s10189-022-00239-6