can be found in the literature for specific polymers can be attributed to such
problems. But, it is reassuring that in a few recent experiments evidence for several
crossovers in hXi versus f curves and for the nonlinear Pincus behavior have
been found [25].
The problem of understanding the persistence length and its consequences is
also taken up by Butt et al. [26]: for bottle-brush polymers, there is the challenging
problem of understanding how their stiffness depends on the grafting density and
degree of polymerization of the grafted side chains.
1.3 Single Chain Collapse Versus Adsorption
The studies on single-chain adsorption on flat substrates were based on the same
models used for the studies of single chains in the bulk, as described above. One
issue that was addressed is the competition between adsorption, collapse, and
crystallization of tethered single chains [5].
Figure 5 presents a tentative “diagram of states” for N ¼ 64 (we should
speak about “phase diagrams” only in the limit N ! ∞, so the lines in the
diagram of states are not sharp phase boundaries, but rather various signatures of
smooth crossovers). Here, we use β b ¼ E/k B T and β s ¼ E s /k B T as control parameters (E and E s denote the strength of the attractive energy between monomers and
between monomers and the substrate surface, respectively). Due to the competition
between the structures identified in the bulk (Fig. 2) and various quasi-twodimensional wall-attached structures, the phase diagram emerging in the limit
N ! ∞ for the bond fluctuation model of a polymer interacting with the surface
(and allowing for variable solvent conditions) is still incompletely understood [5].
When we restrict attention only to the good solvent case, the PERM [17, 18]
algorithm applied to the simple SAW model on the simple cubic lattice can be used
again, and very long chains can be simulated (see Sect. 1.2). Although adsorption of
single flexible polymers under good solvent conditions is a classical problem that
has been studied for decades [27–32], very important aspects are still controversial.
One such aspect is the value of the crossover exponent ϕ, which controls the
number of adsorbed monomeric units N s right at the adsorption threshold. Although
ϕ ¼ 1/2 for Gaussian chains is well known, de Gennes [33] suggested a scaling
relation ϕ ¼ 1 À ν, which would imply ϕ % 0.41. However, then it was shown that
this scaling relation should hold only for a chain tethered to a freely penetrable
interface, but not to an impenetrable surface [27, 28]. Early simulations [27] gave
ϕ % 0.59, but later investigations came up with different values; the smallest
estimate so far is Grassberger’s [30] estimate of ϕ % 0.48, but there has not yet
been any consensus on a value of this exponent. Bhattacharya et al. [31] observed
that, depending on the degree of interaction between different loops, one could get
any value in the range 0.39 ϕ 0.59. Should one draw then the conclusion that
this exponent is nonuniversal?
To test this question, the SAW tethered to an impenetrable wall has been studied,
using a square well adsorption potential of depth U and range W. If universality
8
R. Berger et al.
problems. But, it is reassuring that in a few recent experiments evidence for several
crossovers in hXi versus f curves and for the nonlinear Pincus behavior have
been found [25].
The problem of understanding the persistence length and its consequences is
also taken up by Butt et al. [26]: for bottle-brush polymers, there is the challenging
problem of understanding how their stiffness depends on the grafting density and
degree of polymerization of the grafted side chains.
1.3 Single Chain Collapse Versus Adsorption
The studies on single-chain adsorption on flat substrates were based on the same
models used for the studies of single chains in the bulk, as described above. One
issue that was addressed is the competition between adsorption, collapse, and
crystallization of tethered single chains [5].
Figure 5 presents a tentative “diagram of states” for N ¼ 64 (we should
speak about “phase diagrams” only in the limit N ! ∞, so the lines in the
diagram of states are not sharp phase boundaries, but rather various signatures of
smooth crossovers). Here, we use β b ¼ E/k B T and β s ¼ E s /k B T as control parameters (E and E s denote the strength of the attractive energy between monomers and
between monomers and the substrate surface, respectively). Due to the competition
between the structures identified in the bulk (Fig. 2) and various quasi-twodimensional wall-attached structures, the phase diagram emerging in the limit
N ! ∞ for the bond fluctuation model of a polymer interacting with the surface
(and allowing for variable solvent conditions) is still incompletely understood [5].
When we restrict attention only to the good solvent case, the PERM [17, 18]
algorithm applied to the simple SAW model on the simple cubic lattice can be used
again, and very long chains can be simulated (see Sect. 1.2). Although adsorption of
single flexible polymers under good solvent conditions is a classical problem that
has been studied for decades [27–32], very important aspects are still controversial.
One such aspect is the value of the crossover exponent ϕ, which controls the
number of adsorbed monomeric units N s right at the adsorption threshold. Although
ϕ ¼ 1/2 for Gaussian chains is well known, de Gennes [33] suggested a scaling
relation ϕ ¼ 1 À ν, which would imply ϕ % 0.41. However, then it was shown that
this scaling relation should hold only for a chain tethered to a freely penetrable
interface, but not to an impenetrable surface [27, 28]. Early simulations [27] gave
ϕ % 0.59, but later investigations came up with different values; the smallest
estimate so far is Grassberger’s [30] estimate of ϕ % 0.48, but there has not yet
been any consensus on a value of this exponent. Bhattacharya et al. [31] observed
that, depending on the degree of interaction between different loops, one could get
any value in the range 0.39 ϕ 0.59. Should one draw then the conclusion that
this exponent is nonuniversal?
To test this question, the SAW tethered to an impenetrable wall has been studied,
using a square well adsorption potential of depth U and range W. If universality
8
R. Berger et al.