Swelling and Collapse of Single Polymer Molecules and Gels Single polymer molecules Coil-globule transition If polymer chains are not ideal, interactions of non-neighboring monomer units ( the so-called volume interactions ) should be taken into account. If these interactions are repulsive, the coil swells with respect to its ideal dimensions. If monomer units attract each other, contraction leads to the “condensation” of polymer chain upon itself with the formation of a ”dense droplet” conformation, which is called a polymer globule. Polymer gels The gel as a whole is actually one giant three-dimensional molecule. Swelling of a gel Repulsive interactions of monomer units Collapse of a gel Attractive interactions of monomer units Advantage of studying gels vs. dilute polymer solutions: possibility of direct visual observation of conformational transitions. Main disadvantage: very slow equilibration in macroscopic gel. For the sample of 1 cm it takes several days. Equilibration time can be diminished by using smaller size L of the sample, one can show that ~ L2 DNA macromolecules DNA coil and globule as revealed by fluorescent microscopy The size of DNA coil and even globule can be of order of the wavelength of visible light (400-700 nm), therefore conformational changes with DNA can be seen by optical microscope. Normally in such experiments for better contrast DNA chain is doped with fluorescent dyes. Single Polymer Chains with Volume Interactions Main polymer chain models for the consideration of volume interactions: 1. Model of beads on a Gaussian filament a Beads of volume on immaterial filament. The connectivity of beads is given by a condition that the the probability distribution for a vector r between neighboring beads is 3 3 2 3r 2 g (r ) exp 2 2 2a 2a Comment: chain connectivity as in the model of beads can be obtained for any chain by selecting the dividing points separated by several persistent lengths and by assigning all the mass of each chain segment to the dividing point. The value of a is the average distance between neighboring beads. It is possible to have a 3 or a 3 The beads interact with the interaction potential U (r ).Typical interaction potential U repulsion due to self-volume of the beads (excluded volume) r Van-der-Waals attraction Note: in principle this potential is renormalized by the presence of the solvent, so its form can be much more complicated. 2. Lattice model Polymer chain is modeled as a random walk on a lattice. The random walk cannot visit each site more than once (the excluded volume condition); the lattice attractive energy is assigned to each pair of neighbor sites visited by two nonconsecutive monomer units of the chain. Concept of -temperature It should be noted that for both models: 1. At high values of T, the ratio / kT << 1 and only repulsion matters. The coil should swell with respect to the ideal dimensions; this phenomenon is called the excluded volume effect. In this case the so-called swelling coefficient of the coil, , is larger than unity: 2 R 2 2 1 R 0 2. At low values of T, the ratio / kT >> 1, and attraction dominates. The coil should shrink and form a condensed globule ( coil-globule transition ). 3. At intermediate values of T, the effect of repulsion and attraction should compensate each other and the coil should adopt ideal-chain ( unperturbed ) size. This happens at the so-called -temperature. The free energy of a coil is the sum of energetic and entropic contributions: F E TS As to the energy E, contrary to the case of the ideal coil, for the present case it is not equal to zero. To write down the expression for E we should recall that the concentration of monomer units inside the polymer coil is very small for long chains. Therefore, we can write the expression for E in terms of the expansion in the powers of n: E NkT ( Bn Cn 2 ) Here B, C, … are second, third, … virial coefficients; these coefficients are responsible for binary, ternary, etc. interactions of monomeric units. These coefficients depend only on the interaction potential between the units, e.g. 3 1 B(T ) 1 exp( U ( r ) kT ) d r 2 From the expression for B(T ): at high T ( kT 1 ) B , with the decrease of T the value of B diminishes, until it reaches zero at some temperature . T-Θ At T : B , where T The characteristic dependence of the second virial coefficient on temperature At the -temperature B 0 F TS chain adopts the conformation of ideal coil. At T repulsion dominates, coil swells due to the excluded volume effect - region of good solvent. At T attraction dominates, coil shrinks into globule - region of poor solvent. Remarks: 1. The fact of complete compensation of interactions at the -point is a specific polymer property (not valid e.g. for gases) connected with low polymer concentration in the coil. 2. Repulsion for T and attraction for T is valid for the usual form of the potential U (r ). For more complicated forms the situation may be reverse, or it may be many -points (see below). The Excluded Volume Problem Let us consider the polymer coil far above the -point, in the good solvent region, and let us calculate the swelling of the coil due to the excluded volume. F E TS NkTBn TS 3N 3kTR 2 NkTB const 3 2 4 R 2 Na For the second term we used the expression for the entropy of a polymer coil expanded up to the size R (derived earlier). Excluded volume repulsion (first term) induces the coil swelling, while entropic elasticity (second term) opposes the swelling. The balance (minimization of F with respect to R) gives the equilibrium coil size. The free energy of a coil is N 3kTR 2 F NkTB 4 3 2 R 2 Na 3 Minimization of F with respect to R : F 0 R kTBN 2 kTR 0 4 2 R Na Therefore, R Ba 2 ) 1 5 N 3 5 a 2 ) 1 5 N 3 5 Also swelling coefficient is 1 5 110 1 3 N 1 N 2a a R The polymeric coil swells essentially due to the excluded volume. In spite of the low polymer concentration in the coil, the swelling is large because of high susceptibility of long polymer chains. Swelling of polymer gels We will consider the gels synthesized in the presence of a large amount of solvent. For such gels entanglements between the gel chains are not important. A swollen polymer gel Since each subchain of a gel swells independently in its own subvolume: 1. The swelling degree of each subchain is equal to the swelling degree of the gel as a whole. 2. The swelling degree of each subchain is the same as the swelling degree of an isolated chain in the good solvent. a) 1 5 3 N 1 10 1 Superabsorbing properties of polyelectrolyte gels In real case the value of varies between 1.2 and 2.5 which corresponds to a 15-fold increase of the volume of the gel upon swelling in the maximum, except for the case of polyelectrolyte gels. counter ions charges on polymer chains Schematic picture of a polyelectrolyte gel Polyelectrolyte gels normally increase their volume in water several hundred times. This property is used for the design of superabsorbing polymer systems. The superabsorbing properties of polyelectrolyte gels in water are quite universal: they are observed for any gel, independently of its chemical structure, provided that the gel contains charged monomer units. Since there are charges on polymer chains, there should also be counter ions, because the overall system should be electrically neutral. For these counter ions it is thermodynamically advantageous to abandon the gel and travel in the whole external volume, because in this way they gain significant entropy of translational motion. However, this is impossible, because this would violate the condition of macroscopic electroneutrality of the gel. The counter ions are therefore forced to be confined within the gel, and they produce a significant osmotic pressure in the gel. This osmotic pressure induces a very essential gel swelling: in this way there is more space for the translation motion of each counter ion. Coil-Globule Transition Now let us consider the whole range of temperatures. When the temperature is lowered below the Θ-point, the coil-globule transition(or polymer chain collapse) should take place. T > ; >1 good solvent T < ; <1 poor solvent The interest to globular form of macromolecules is induced by molecular biophysics: proteins-enzymes are polymeric globules. Denaturation of proteins is sometimes considered to be analogous to the transition from globule to coil. To determine the characteristics of the coil-globule transition, let us write F E TS, as before, but now the expression for both E and S should be different. Energy: For dense globules higher virial coefficients may be of importance: E NkT Bn Cn 2 ) The term Bn gives the attraction at T , while the term Cn2 gives repulsion, since normally C 2 > 0 in the region of interest. The higher-order virial coefficients can be neglected in the coil-globule transition region. Thus, omitting, as usual, all numerical coefficients, we have BN CN 2 E NkT 3 6 R R Entropy: We must take into account the possibility of chain compression (not only extension). 1 In case of coil R N 2 a 3kR 2 R2 S 2 2 2 Na Na For the globular state 1 R N 2 a N Na 2 S k k 2 g R The subchain of g links with ga2 R2 is practically free ( ~ one encounter with the “wall” loss of entropy 1 per this 2 2 subchain); g R a . Interpolation formula valid (in the orders 1 of magnitude) for R N 12 a , R N 2 a and in the intermediate region: R 2 Na 2 S k 2 2 R Na Finally we have for the free energy: F E TS BN CN 2 R 2 Na 2 NkT 3 6 kT 2 2 R R R Na Minimization of F with respect to R ( F R 0 ) gives 3 R5 CN 2 2 RNa BN Na 2 R3 2 C υ 2 2 2 Or in terms of R Na , y 6 6 , a a 1 1 2 BN υτ N 2 x 3 a a3 we have 5 y 3 x 1 1 1 2 3 2 3 y=1 y = 1/ 60 y = 1/ 300 x -2 poor solvent -1 0 - point 1 2 good solvent Conclusions: 1 Coil-globule transition takes place at x 1, 1 1 3 3 2 i.e. N a 1 a N 2 1, this is only slightly lower the -point. It is enough to have a very weak attraction to induce the transition into globule (this is not the case for condensation of gases). Reason: due to the chain connectivity the independent motion of monomer units is impossible (polymer coil is poor in entropy). 2 For y<<1 the chain collapse is discrete, while for y ~ 1 it is continuous. The value of y depends on /a3 ( y 2/a6 ). For << a3 y << 1, while for a3 y ~ 1. For realistic chain models y << 1 corresponds to stiff chains, and y 1 to flexible chains. Indeed, it can be shown that C d 3 l 3; a l , therefore y = C / a6 d 3/ l 3. Thus, for stiff polymer chains the collapse is discrete, while for flexible chains it is continuous. ) 3 In the limit of small , B < 0 ( globular region ) the size of the globule is defined by the following two terms: CN 3 3 BN 2 R Thus the size of the globule is CN R B 3 or R C B ) N 1 3 1 3 The volume fraction of monomer units within the globule is nυ B υυ τ N υ 2 τ 3 R C υ 1 (a) For the globule R ~ N 3 (cf. with 3 1 R ~ N 2 for ideal coil and R ~ N 5 for the coil with excluded volume). 3 (b) In the globule far from -point ( || 1) the volume fraction of monomer units is generally not small. This is a dense liquid droplet. 3 (c) The globule swells as the -point (and the coil-globule transition point) is approached, so the description in terms of B and C only in the vicinity of transition point is valid. Φ Θ τ The volume fraction of 4 Experimentally the coil-globule transition was observed for many polymer-solvent systems. A very convenient system is polystyrene in cyclohexane, since the -temperature for this case corresponds to = 35 oC. Main difficulty for the experimental observation of the coil-globule transition is connected with the possibility of intermolecular aggregation and formation of precipitate, rather than coil-globule transition. To avoid this the concentration of polymer in the solution should be very small ( e.g. for polystyrene-cyclohexane system it should be less than 10-4 g/l ). Formation of precipitate in the poor solvent region 5 One of the most interesting possibilities to study the coil-globule transition for the particular case of DNA molecules is to use direct visualization via fluorescence optical microscopy. In this case one may work in the range of much lower polymer concentrations (up to 10-5 g/l) and the problem of chain aggregation is not very serious. One may observe the bimodality of the histogram of observed sizes which is the indication of the first ordered coil-globule transition. This is in full agreement with the theory outlined above, since DNA is a stiff-chain polymer. Histogram of DNA sizes as a function of concentration of added poly(ethylene oxide) - agent making solvent quality poorer. Collapse of polyelectrolyte gels At some critical solvent composition the several-hundred-fold jumpwise decrease of the gel volume is observed. V/V0 1,0 The dependence of a volume of a polyelectrolyte gel V on the volume fraction of the poor solvent added to water. 0,8 0,6 0,4 0,2 0,0 0 20 40 60 80 100 , % The amplitude of the jump is directly connected with the polyelectrolyte nature of the gel: it increases with an increase of the degree of charging of the gel chains. The gel shows high responsiveness to the external actions in the region of jumpwise collapse. In this region it is enough to change the external conditions ( solvent quality, salt concentration, pH, temperature ) only slightly to obtain a very strong reaction of the system.