Combination of SEC/MALS experimental procedures and theoretical analysis for studying the solution...

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J. Biochem. Biophys. Methods 56 (2003) 95–116

Combination of SEC/MALS experimental

procedures and theoretical analysis for studying

the solution properties of macromolecules

M. Pilar Tarazona*, Enrique Saiz

Departamento de Quımica Fısica, Universidad de Alcala, 28871 Alcala de Henares, Spain

Abstract

Size-exclusion chromatography (SEC) with dual detection, i.e., employing refractive index (RI)

and multiangle light-scattering (MALS) detectors, has been applied to study the solution properties

of two very different polymer–solvent systems at 25 jC: poly(N-vinylcarbazole) (PVCz) in an

organic solvent THF that is a very good solvent and a system under h conditions that is formed by

polyvinylpyrrolidone (PVP) in water containing a 0.1 M concentration of NaNO3. In both cases, the

analysis of a single highly polydisperse sample obtained by free radical polymerization is enough for

obtaining molecular weight and radius of gyration calibration curves, molecular weight distributions

(MWD) (and thus, molecular weight averages), molecular dimensions, scaling laws coefficients and

unperturbed dimensions. Extrapolation to h conditions produces values of the characteristic ratio of

the unperturbed dimensions Cn = hr2io/nl2 = 15.9 and 14, respectively, for PVCz and PVP.

Unperturbed dimensions are also theoretically calculated with different models such as Kuhn

equivalent chain, worm-like chain and rotational isomeric states model. Conformational parameters

required for this last model were taken from the literature in the case of PVCz; however, they are

calculated by molecular dynamics simulations in the case of PVP. Theoretical values obtained with

the RIS model are in good agreement with the experimental results.

D 2003 Elsevier Science B.V. All rights reserved.

Keywords: Size-exclusion chromatography; Solution properties; Molecular dimension; Conformational proper-

ties; Poly(N-vinylcarbazole); Polyvinylpyrrolidone; Molecular dynamics

1. Introduction

Size-exclusion chromatography (SEC), coupled with molecular weight-sensitive detec-

tors, is the most widely used technique for determining the absolute molecular weight

0165-022X/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0165-022X(03)00075-7

* Corresponding author.

E-mail address: [email protected] (M.P. Tarazona).

M.P. Tarazona, E. Saiz / J. Biochem. Biophys. Methods 56 (2003) 95–11696

distribution (MWD) and averaged molecular weights of synthetic and natural polymers

[1–3]. Light-scattering detectors are very useful since they provided absolute measure-

ment of molecular weight without requiring any calibration. Besides, these detectors are

very efficient, detecting aggregates even at low concentrations. Multiangle laser light

scattering (MALS) offers an added advantage: the determination of the molecular

dimensions of the polymer, in terms of the radius of gyration, from the angular dependence

of the scattered light [4,5]. This combination of molecular weight and dimensions of the

polymer chains can be used to obtain important information about size, shape and

conformation of polymer chains in solution [6].

A macromolecule can adopt many different conformations by rotation over the covalent

single bonds that form the chain skeleton [7]. In most circumstances, the interconversion

from one to another conformation is very easy, and the polymer changes among the

allowed conformations. Molecules that are moving among many different allowed

conformations are called random coils. Typical examples of random coil molecules are

synthetic polymers in solution or amorphous bulk state above its glass transition temper-

ature Tg. When the transit from one to another conformation becomes impossible, the

molecule stays ‘‘frozen’’ in one conformation and exhibits very different properties than

when it was in the random coil state. Examples of fixed conformations are polymers at low

temperature (below Tg), in crystalline state or biopolymers in native state.

In very diluted solutions, the intermolecular polymer–polymer interactions are negli-

gible, and the behavior of the sample is dictated by the balance among intramolecular

polymer–polymer and intermolecular polymer–solute interactions, which strongly depend

on the thermodynamic quality of the solvent. Thus, in the so-called good solvents, the

polymer–solvent is more favorable that the polymer–polymer, the polymeric chain tends to

solvate, minimizes the number of polymer–polymer contacts and produces a rather large

excluded volume. Extended conformations are therefore favored, and the polymer exhibits

high molecular dimensions. On the contrary, in poor solvents, the polymer–polymer

interactions are preferred over the polymer–solvent ones; the polymer units try to avoid

contact with the solvent, and the number of polymer–polymer contacts increases, producing

a collapse of the chain that gives rise to a negative value of the excluded volume. Compact

conformations are thus favored, and the polymer has much smaller molecular dimension

than in the previous case. When the quality of the solvent decreases below a certain limit, the

self-avoiding process among polymer and solute comes the point of producing phase

separation, and the polymer precipitates. Avery interesting intermediate situation is reached

when the polymer–solvent interactions exactly counterbalance the so-called long-range

polymer–polymer interactions, i.e., all intramolecular interactions except those taking place

among immediate neighbor units. Polymer–polymer and polymer–solvent contacts are then

equivalent as far as the interactions are concerned, the excluded volume vanishes and the

system is said to be in the unperturbed conditions or h (theta) conditions [8]. The molecular

dimensions are then intermediate among those obtained in good and in bad solvents.

Furthermore, the unperturbed conditions represent a very convenient situation for theoretical

calculations since the probability of existence of any given conformation depends only on a

few short-range interactions among neighbor units that are relatively easy to compute. This

is one of the reasons why experimental results are often measured or extrapolated to

unperturbed conditions in order to facilitate the comparison with theoretical values.

M.P. Tarazona, E. Saiz / J. Biochem. Biophys. Methods 56 (2003) 95–116 97

The radius of gyration hs2i is a very convenient magnitude to express the molecular

dimensions of a polymer chain because, in the first place, it can be directly obtained from

light-scattering data, and its definition is applicable to macromolecules of any shape

[7,9]. Furthermore, the value of hs2i for a given body, a polymer chain for instance, is

related to its mass by simple mathematical relationships, called scaling laws, whose

numerical parameters depend on the shape of the body. Thus, the analysis of exper-

imental values of hs2i as a function of molecular weight affords information on the shape

of the polymer chain.

Synthetic polymer samples are usually polydisperse, and therefore, the molecular

weights obtained with different techniques are different averages to which the scaling laws

can not be applied. It is in this context where the combination of SEC with MALS

becomes a very convenient procedure since it can analyze a highly polydisperse sample

providing simultaneous values of radius of gyration and molecular weight of the polymer

chains that abandon the column, which, provided that the efficiency of the separation

process is good, are practically monodisperse. Therefore, a single polydisperse sample

suffices for a full analysis that, on the other hand, becomes more accurate since it does not

depend on problematic fractionation procedures [10].

In the present work, we apply this technique to the conformational analysis of two

different polymer solutions. The first one was poly(N-vinylcarbazole) PVCz in THF at 25

jC, which is an example of polymer in a good solvent. The second solution was formed by

poly(N-vinylpyrrolidone) PVP in water containing a 0.1 M concentration of NaNO3 at the

same temperature of 25 jC at which the system is below (although close to) unperturbed

conditions. In both cases, a single polydisperse sample obtained by free radical polymer-

ization was enough for obtaining molecular weight averages and distributions, molecular

dimensions, coefficients for the scaling law, etc.

Unperturbed values of the molecular dimensions were analyzed in terms of two very

idealized theoretical models, namely, the Kuhn equivalent chain and the worm-like chain.

A more realistic calculation employing a rotational isomeric states model, whose

parameters depend on the chemical structure of the polymer, was also performed and

the results compared with the experimental values in order to gain a deeper understanding

of the shape and behavior of the samples.

2. Materials and methods

2.1. Measurements

Size-exclusion chromatography measurements were carried out using a Waters Asso-

ciates equipment with a model 510 pump, a U6K injector and a differential refractive

index (RI) detector model 410 (RI). The multiangle light-scattering (MALS) detector was

a DAWN DSP-F laser photometer from Wyatt Technology. The photometer was calibrated

with spectrometric-grade toluene (Scharlau). The normalization of its detectors in THF and

aqueous solution was performed with low-molecular-weight standard samples of poly-

styrene and dextran, respectively. The same polystyrene standard was used to determine

the interdetector volumes [4].

2.2. Materials

For organic SEC, two columns PLgel mixed B (Polymer Laboratories) in series

completed the equipment. The polymer PVCz was synthesized by radical polymerization

[11] and THF freshly distilled from sodium and benzophenone, filtered through a 0.2-AmFluoropore membrane and degassed, was used as eluent. The flow rate was 1.0 ml/min and

the temperature 25 jC. For aqueous SEC, two Ultrahydrogel Linear 6–13 Am were the

chromatographic columns, the eluent was a 0.1 M NaNO3 aqueous solution filtered

through a 0.2-Am cellulose membrane and degassed. The polymer PVP was a commercial

sample (Aldrich). The flow rate was 0.5 ml/min and the temperature 25 jC.

2.3. Coupled SEC and MALS

The basic light-scattering equation is [12]

Kc

DRh¼ 1þ 16p2

3k2hs2isin2 h

2

� �þ . . .

� �1

Mw

þ 2A2cþ . . .

� �ð1Þ

where DRh is the excess Rayleigh ratio, c is the concentration, k is the wavelength of the

incident light in the medium, h is the scattering angle and K is the optical constant given by

K ¼ 4p2n2

k40NA

dn

dc

� �2

ð2Þ

where k0 is the vacuum wavelength, n is the solvent refractive index, NA is Avogadro’s

number and dn/dc is the refractive index increment in the polymer solution.

The first parenthesis of Eq. (1) contains the terms of the particle form factor that take

into account the interferences among light scattered at different points within large

molecules, whereas the second parenthesis includes the nonideal terms arising from the

polymer–polymer interactions. Since the concentration used in SEC is very small (usually

1–5 mg/ml) and the samples are even more diluted through the column, the term 2A2c is

negligible, and thus, Eq. 1 simplifies to

Kc

DRh¼ 1

Mw

1þ 16p2

3k2hs2isin2 h

2

� �þ . . .

� �ð3Þ

The RI detector measures the concentration, whereas MALS detectors measure

simultaneously the light scattered at several angles, and a plot of Kc/DRh versus

sin2(h/2), Zimm plot, affords to calculate the molecular weight from the intercept and

the radius of gyration hs2i from the slope.

When a polydisperse polymer sample is used, the different fractions eluted (slices in the

chromatogram) are assumed to be of a single molecular weight [4]. Thus, M, hs2i and c aresimultaneously measured for each slice that elutes at volume vE. These data can be used to

obtain the calibration curves (M or hs2i versus vE), the molecular weight distribution,

MWD, and the scaling law of the polymer.



3. Results

3.1. Chromatograms

Fig. 1 shows the RI and one of the MALS signals (at 90j) for a sample of PVP. The

form of the signals is different since the RI signal is proportional to c, whereas the light-

scattering intensity is proportional to the product Mc (Eq. (3)). The corresponding

calibration curves of molecular weight and radius of gyration versus elution volume are

represented by open and filled circles, respectively, in Fig. 1. The dispersion of the points

for values of radius of gyration lower than 20 nm (elution volumes higher than 14 ml) is

due to the increase of the error in the determination of this magnitude as the molecules

approximate the isotropic scatter behavior and the angular dependence of scattered light

diminishes.

Fig. 2 depicts the molecular weight distribution for the same sample PVP. The different

averaged molecular weights obtained with the ASTRA software are Mn = 5.2� 105,

Mw = 1.14� 106 and Mz = 2.15� 106, and the polydispersity of the sample Mw/Mn = 2.2.

As can be seen in Figs. 1 and 2, this sample is broad enough as to offer a collection of

molecular weights and radii of gyration that can be used to obtain information about its

solution properties.

3.2. Scaling laws

The scaling law relating the size and the mass of a macromolecule can be expressed as

hs2i1/2 =QMq [7,9,13]. The scaling law exponent q depends on the conformations of the

Fig. 1. Calibration curves of molecular weight (o) and radius of gyration (.) versus elution volume for a sample

of PVP. The corresponding RI and one of the MALS signals (at 90j), in arbitrary units, are also depicted.

Fig. 2. Molecular weight distribution for PVP.


molecule and the thermodynamic quality of the solvent. For h conditions, q = 0.5 and the

unperturbed radius of gyration hs2io scales with M. The exponent q increases with the

quality of the solvent and reaches the value 0.6 for very good solvents. Below the theta

conditions, the chains shrink, avoiding the solvent molecules, q decreases and, in the limit,

the polymer assumes a globular shape for which q = 1/3.

Fig. 3 shows the log–log plots of the radius of gyration versus molecular weight for

PVP in 0.1 M NaNO3 and PVCz in THF. For PVCz, the plot is linear in the whole range of

Fig. 3. Scaling laws for PVCz (o) and PVP (.).


molecular weights, and the slope provides q = 0.61F 0.01, indicating that the polymer

behaves as a random coil in a very good solvent. For PVP, the plot shows a slight

curvature at low molecular weights. The slope of the linear part yields q= 0.40F 0.02,

indicating that the polymer chains are random coils under h conditions.

3.3. Unperturbed dimensions for PVCz

As explained above, the dimensions of a polymer in a good solvent are perturbed due to

the excluded volume effects and can be related to the unperturbed dimensions through the

chain expansion factor, a:

hs2i ¼ a2hs2io ð4Þ

The expansion factor depends on the polymer chain, the quality of the solvent and the

molecular weight. The expansion factor is unity in two circumstances: first, at hconditions, when the dimensions of the chain are the unperturbed dimensions, and second,

at the limit of such a low molecular weight that the interaction between segments of the

same chain is not possible. The dependence of the expansion factor on molecular weightM

can be used to extrapolate results obtained in perturbed condition to h conditions [10].

However, the dependence is complex, and several theoretical and empirical expressions

can be founded in the literature. For polymers in good solvents, a widely used expression

is the Stockmayer–Fixman [14]

a3 ¼ 1þ CM 1=2 ð5Þ

The combination of Eqs. (4) and (5) affords:

hs2iM

� �3=2

¼ hs2ioM

� �3=2

ð1þ CM 1=2Þ ð6Þ

This equation is similar to the well-known extrapolation of Stockmayer–Fixman [14]

for viscosity measurements, and the intercept provides (hs2io/M)3/2.

The Fixman [15] extrapolation is defined in Eq. (7), and provides hs2io/M as the

intercept:

hs2iM

¼ hs2ioM

þ 0:0299Bhs2ioM

� ��1=2

M 1=2 ð7Þ

The last equation used in this work is due to Kurata et al. [16]:

hs2iM

¼ hs2ioM

þ 0:0286BgðaÞ M

hs2i1=2

!ð8Þ

The three plots are represented in Fig. 4. There is good concordance between the three

extrapolations, and the mean value of hs2io/M is 6.5F 0.1�10� 4 nm2 mol g� 1.

Fig. 4. Different extrapolations to unperturbed dimensions for PVCz. Fixman (squares), Stockmayer–Fixman

(triangles) and Kurata–Stockmayer–Roig (circles).


3.4. Unperturbed dimensions for PVP

The scaling law for PVP indicates that the polymer is under theta conditions, and the

extrapolations used for PVCz are not valid. Thus, a different procedure must be employed

[17,18]. For polymers close to the h conditions, the dependence of a on molecular weight

is better expressed according to the two parameters theory [9]:

a2 ¼ ð1þ 1:276z� 2:082z2 þ . . .Þ ð9Þ

where z is a parameter proportional to M1/2 and to the excluded volume interaction energy

of the polymer system, which is negative if the system is under theta conditions. Thus, the

values of hs2i/M versus M1/2 should fit a relationship:

hs2iM

¼ hs2ioM

ð1þ AM 1=2 þ BM þ . . .Þ ð10Þ

and with the omission of the z2 and higher terms, the relationship is linear, with a negative

slope and hs2io/M as the independent term. Fig. 5 shows the experimental values of hs2i/Mversus M1/2 for PVP in NaNO3 0.1 M. The upwards curvature exhibited by the

extrapolation is a straight consequence of under theta conditions of the system. Thus, the

value of a should decrease below unity, and the ratio hs2i/M will increase with decreasing

M. Linear fit of the same points used for obtaining the scaling law is also drawn. The value

of the slope is � 8.1F 0.1�10� 8, negative and very small, as corresponds to near hconditions, and the intercept affords hs2io/M = 1.0F 0.1 nm2 mol g� 1.

Fig. 5. Extrapolation to unperturbed dimensions for PVP.


3.5. Characteristic ratio

The ratio hs2io/M is independent of molecular weight, and thus, it could be used as a

characteristic parameter of a given polymer chain. However, other related magnitudes such

as characteristic ratio or persistence length are more frequently used for characterizing the

chain. The characteristic ratio Cn is defined by the ratio of the unperturbed dimensions to

the dimensions of a freely jointed chain and can be calculated from the extrapolated values

of hs2io/M as [7]:

Cn ¼hr2ionl2

¼ 6Mo

2l2hs2ioM

ð11Þ

where n is the number of bonds, each one of them having a length l = 0.154 nm in the

present work, hr2io the unperturbed value of the mean square end-to-end distance and Mo

the molecular weight of the repeat unit. The hr2io = 6hs2io relationship valid for flexible

chains was used in the above equation. The value of Cn obtained for PVCz (Table 1) is in

very good agreement with the values for different solvents reported in the literature, which

Table 1

Parameters for the Kuhn equivalent chain (N and L) and the worm-like chain (rmax and a) obtained from the

experimental values of Cn

Parameter PVCz PVP

Cn (experimental) 15.9 14

N/n 0.043 0.049

L (nm) 2.95 2.60

rmax/n (nm) 0.128 0.128

a (nm) 1.48 1.30

All the valence angles of the chain were assumed to be 112j for the calculation of rmax.


range from 15 to 16 [19]. For PVP, the value obtained is slightly higher than that obtained

for viscosity measurements in water, 12.3 [19]. However, the unperturbed dimensions of

this polymer strongly depend on the solvent used [20].

4. Theoretical analysis

The molecular dimensions of a polymeric chain can also be theoretically calculated in

terms of molecular parameters such as bond lengths, bond angles, rotational states and

conformational energies, see Fig. 6. Unfortunately, a polymer chain is too complicated as

to allow a fully realistic calculation so that any theoretical approach is based on the use of

models that represent the actual chain in a more or less idealized manner.

4.1. Simple models

4.1.1. The freely jointed chain

The simplest of all models is the freely jointed chain [7,21] that represents the

polymeric chain by a set of n bonds, each one of them having a length l (both n and l

being the actual values on the real chain), joined in linear succession, the angles at the

bond junctions being free to assume all values with equal probability and the rotation

about bonds being likewise free. It is trivial to prove that, with these assumptions, the

averaged molecular dimension are given by:

hr2io ¼ 6hs2io ¼ nl2 ð12Þ

The result indicates that the square of the unperturbed molecular dimensions scale with

n, or with the molecular weight, as measured values in fact do. However, actual values of

hr2io or hs2io are noticeably larger than those obtained with this simple model, indicating

that the real polymers are more rigid than what the model assumes. In fact, this model is

employed as reference, and thus, the characteristic ratio Cn measures the discrepancy

between the actual and idealized dimensions.

4.1.2. The Kuhn equivalent chain

However, since the freely jointed chain is a very simple model that employs only one

structural parameter (namely, the bond length l), besides the number of skeletal bonds,

Fig. 6. A rough sketch of a polymeric chain shown in its all trans conformation. Skeletal atoms are numbered

from 1 through n+ 1. Skeletal bonds are numbered from 1 to n so that bond i links atoms i� 1 and i. Rotational

angle /i is defined as the dihedral angle formed by the planes defined by bonds i� 1,i with i,i + 1. A set of values

for all rotational angles excluding the first and last ones, i.e., {/2, /3,. . ., /n� 1}, is referred to as a conformation

of the chain.


which is proportional to the molecular weight, it is also very appealing for the

representation of real macromolecules, even if the parameters have to be adjusted in

order to reproduce the actual chain dimensions. The procedure employed was devised by

Kuhn [22] and consists of replacing the actual number n and length l of the skeletal bonds

by adjusted values N and L computed in such a way that, when employed within the

model, will reproduce the actual values of square averaged and maximum end-to-end

distances of the polymer, i.e.:

hr2io ¼ NL2 rmax ¼ NL ð13Þand therefore:

L ¼ hr2iormax

N ¼ r2max

hr2ioð14Þ

The maximum length rmax is customarily referred to as length of the fully extended

chain or contour length, and it is taken to be the maximum achievable length without

unphysical distortion of the molecular geometry [23], which is often identified with the

length of the all-trans conformation. Assuming that all the skeletal bond angles have the

same value h, rmax = nlsin(h/2). On the other hand, the value of hr2io can be written as

hr2io =Cnnl2, and therefore, Eq. (14) may be expressed as:

L ¼ Cn

sinðh=2Þ l N ¼ ½sinðh=2Þ2

Cn

n ð15Þ

so that L>l and N < n, i.e., fewer bonds but longer than the real ones are required to

reproduce the actual dimensions of the chain.

A freely jointed chain containing N bonds of length L is often referred to as equivalent

to the real chain of n bonds with length l. Table 1 shows the values of L and the ratios N/n

calculated according to Eq. (15) from the experimental results of Cn for the polymers

studied in this work.

4.1.3. The freely rotating chain

The freely jointed chain model can be improved by incorporation of real features of the

polymer chain [17,21], the most obvious one being a fixed value for the bond angle h.Thus, the freely rotating chain model represents the polymer by a set of n bonds of length l

joined successively with a constant value h of the angle at the junctions. Rotational angles

/, however, may adopt any value from 0j to 360j with the same probability. Again, it is

very easy to prove that for this model:

hr2io ¼ nl21þ a1� a

� 2að1� anÞnð1� aÞn

� �cnl2

1þ a1� a

Cn ¼1þ a1� a

� 2að1� anÞnð1� aÞn c

1þ a1� a

ð16Þ

with a = cos(p� h) =� cos(h). The approximations in Eq. (16) represent the asymptotic

limits for large values of n. In many polymers, hc 112j, ac 0.37 and the model gives

Cnc 2.2.

4.1.4. The worm-like chain

A limiting case of the freely rotating chain model is the Porod–Kratky [24,25] chain,

which is frequently referred to as worm-like or continuously curved chain. Let us imagine

an hypothetical freely rotating chain containing N bonds of length L. The averaged

projection of bond k over the direction of the first bond is given by hxki = Lak� 1, and

therefore, the averaged projection of the end-to-end distance r that joins the first and last

skeletal atoms may be calculated as:

hrxi ¼XNk¼1

hxki ¼ lXN�1

k¼0

ak ð17Þ

The limiting value of hrxi for an infinitely long chain (i.e., N!l) is called persistence

length a:

au limN!l

hrxi ¼Xlk¼1

hxki ¼ LXlk¼0

ak ¼ L

1� að18Þ

The definition of persistence length as the projection of r over the direction of the first

bond is independent of the model. However, the value L/(1-a) given in Eq. (18) applies

only to freely rotating chains. In a real polymer, a increases with the stiffness of the chain,

and it is independent of molecular weight (or number of bonds n) since it has to be

extrapolated to an infinitely long chain.

The worm-like model is obtained through extrapolation to N!l, L! 0 and h! 180

(i.e., a! 1) in such a way that both the persistence and contour lengths (a and rmax)

maintain the same values that they have in the real chain. Under these conditions

[7,21,24,25]:

hr2io ¼ 2armax 1� a

rmax

1� exp � rmax

a

� h i� ð19Þ

which provides the molecular dimensions in terms of two parameters, namely, the

persistence length characterizing the stiffness of the chain and the contour length which

is proportional to the molecular weight.

Eq. (19) may be simplified in two limiting cases that correspond to rmax/aH1 and rmax/

ab1. The first situation represents a chain that is very long (high value of rmax), very

flexible (low value of a) or both. Under these circumstances, both the exponential and the

ratio a/rmax appearing on Eq. (19) are negligible, and one obtains:

hr2io ¼ 2armax Cn

2armax

nl2¼ 2a

lsinðh=2Þ ð20Þ

Most synthetic polymers fall into this situation which is often called the Gaussian limit.

Comparison of Eqs. (20) and (15) shows that L= 2a, i.e., the Kuhn equivalent segment is

twice the persistence length.



The second limit corresponds to chains being very short (low value of rmax), very rigid

(high a) or both, and it is called rigid rod limit. Under these conditions, the exponential

appearing in Eq. (19) can be expanded in series, and the equation simplifies to:

hr2io ¼ r2max Cn ¼r2max

nl2¼ nsin2ðh=2Þ ð21Þ

The last two rows on Table 1 show the values of contour length rmax and persistence

length a obtained from the experimental values of Cn of the polymers studied in this work.

These values of rmax and a were then used to compute hr2io according to Eq. (19). The

results are shown in Fig. 7 by the ratio hr2io/nl2 as a function of n which tends

asymptotically to the experimental value of Cn.

It is important to realize that even if the calculated values of Cn are in excellent

agreement with the experimental results, neither the Kuhn equivalent chain nor the worm-

like model could be considered realistic microscopic models, since their parameters are

obtained by fitting of experimental results.

4.2. Realistic models. Rotational isomeric states

Any more realistic representation of a polymeric chain requires handling discrete states

for the rotational angles / over single skeletal bonds (the so-called rotational isomeric

Fig. 7. Characteristic ratio of the unperturbed dimensions Cn= hr2i0/nl2 calculated for PVCz and PVP chains

according to the worm-like model as a function of the number of skeletal bonds. Values of the persistence length

were obtained from the experimental results of Cn according to Eq. (21) and then employed to compute hr2i0 withEq. (20).


states [7,21,26–29]) with probabilities dictated by Boltzmann factors of their correspond-

ing conformational energy. Thus, the probability of finding the chain in a conformation cdefined by a set of values of the rotational angles {/2, /3,. . ./n� 1} is given by:

Pc ¼ Z�1exp � Eð/2 . . ./n�1ÞkT

� �ð22Þ

where E(/2. . . /n� 1) represents the potential energy of the polymeric chain in the cconformation and Z represents the partition function, i.e., the sum of Boltzmann’s factors

for all the allowed conformations.

Unfortunately, it is impossible to compute the probabilities of all the allowed

conformations according to Eq. (22), and consequently, some approximations have to

be used, the most frequent one being to write the energy as a sum of contributions:

Eð/2 . . ./n�1Þ ¼ E0 þ E1 þ E2 þ . . .þ En�2 ð23Þ

where Ek is called k-order interaction and represents the sum of all contributions to the

conformational energy that depend on the simultaneous orientation of k consecutive

rotational angles. Thus, for instance, E0 contains contributions such as bond-stretching or

angle-bending energies which do not depend on any rotational angle and therefore is a

constant term for all the conformations. On the contrary, the interaction between first and

last skeletal atoms depends on all rotatable bonds and consequently will be one of the

contributions to En� 2.

The approximation consists of truncating the sum given by Eq. (23) beyond a certain

order, i.e., taking Ek= 0 for k larger than a predetermined value, t. Of course, as the

truncation value t increases, the computed energy approaches its true value given by Eq.

(23), and therefore, the model becomes more realistic. Unfortunately, the complexity of the

model also increases (and rather fast) with the value of t. For instance, taking t= 0, i.e.,

assuming than the only nonzero term in Eq. (23) is E0, reverts to the freely rotating chain,

since all the conformations will have the same energy and, consequently, the same

probability.

A value t= 1 (only zero- and first-order interactions are taken into account) produces a

model customarily called chain with restricted and uncorrelated rotations. This model is

slightly more realistic than the freely rotating chain, although it is not good enough for

most applications.

Probably the best balance between complication and realism of the model [7,21,26–30]

is attained with t = 2 that disregards interactions of order higher than 2 (the so-called long-

range interactions). The resulting model is called chain with restricted and neighbors-

correlated rotations, and it is by far the most frequently used for realistic calculations. It is

important to realize that intramolecular long-range interactions appearing in Eq. (23) are

compensated by intermolecular interactions with a theta-solvent or with other chains in the

amorphous bulk state. Consequently, this model provides an excellent approximation for

polymers in unperturbed conditions.

Assuming that long-range interactions are neglected, the polymeric chain may be

treated as a set of n pairs of bonds whose rotations are correlated but independent from

those of all other pairs. The procedure employed to develop the model is usually called


matrix multiplication scheme [7,21,26–30] because it consists of applying the rules of

matrix multiplication to compute any desired property of the chain. Thus, assuming that

each skeletal bond of the chain has m allowed states (frequently the three staggered ones,

namely trans, gauche and negative gauche) represented by a, b,. . ., m, the statistical

weights of each allowed conformation for the pair of bonds i� 1,i may be written as a

matrix of dimensions v� m whose general element ucv = exp(�Ecv/kT) is the Boltzmann

factor of the energy (up to second-order interactions only) of the conformation obtained

when bond i� 1 is in rotational state c and bond i is in the v. These energies and their

corresponding statistical weight may be calculated for model compounds having the

structure of a diad, or a small oligomer, of the real chain employing molecular mechanics,

molecular dynamics, quantum mechanics, etc. However, sometimes they are also treated

as adjustable parameters to be optimized by comparison between calculated and measured

values of some properties of the polymer.

Once the U statistical weight matrices for all distinct pairs of bonds within the chain are

known, the conformational partition function of a chain containing n bonds is obtained as:

Z ¼ U1

Yn�1

i¼2

Ui

" #Un ð24Þ

where U1 is a row of one unity followed by m� 1 zeros and Un is a column of v unities.

On a similar way, the probability for the pair of bonds i� 1,i within a chain containing

n bonds being in rotational states cv may be computed as:

Pcv ¼ Z�1U1

Yi�1

k¼2

Uk

" #U 0

i

Yn�1

k¼iþ1

Uk

" #Un ð25Þ

where Ui0 is obtained from matrix Ui by replacing all its elements by zero except element

ucv which is left unchanged.

Most of the properties of the polymer chain in a given conformation can be calculated

by the serial product of generator matrices constructed in such a way that their product

generates all the terms required to compute the desired magnitude [7,21,26–30]. Statistical

averages over all the allowed conformations may be obtained by combining the generator

and statistical weight matrices to produce supermatrices whose serial product performs all

the required averages. Alternatively, Eq. (25) may be used to compute the a priori

probability for all the allowed orientations of all pairs of bonds contained in the polymeric

chain. Monte Carlo procedures are then employed to generate conformations of the chain

according to those probabilities. The desired magnitude is computed for each one of the

generated conformations, and the results are finally averaged.

In the case of vinyl polymers, three rotational isomers are usually allowed, i.e., trans,

gauche and negative gauche. However, when the side group is bulky and roughly planar,

as it occurs in the cases of PVCz and PVP, all the conformations containing negative

gauche orientations produce severe seteric repulsions that precludes those conformations

[31]. On the other hand, three different kinds of pairs of bonds have to be considered

within these chains, see Fig. 8. The first kind is the interdiad CH2–CHR–CH2 pairs of

bonds, i.e., bonds meeting at the CHR group such as i� 2,i� 1, i,i + 1 or i + 2,i+ 3 in Fig.

8 The statistical weight matrices for this kind of pair are customarily represented by UV.

Fig. 8. A rough sketch of a meso-racemic triad of PVCz or PVP chains illustrating the three kinds of pairs of

bonds contained in those chains, namely, CH2–CHR–CH2 (such as i� 2,i� 1; i,i + 1 or i+ 2,i + 3), CHR–CH2–

CHR in meso diads (such as i� 1,i) and CHR–CH2–CHR in racemic diads (such as i + 1,i + 2).


The second kind is the pair of bonds CHR–CH2–CHR in meso diads, i.e., pairs of bonds

meeting at the CH2 group within a meso diad such as i� 1,i in Fig. 8. The corresponding

statistical weight matrices are represented by Um. Finally, pairs of bonds CHR–CH2–

CHR in racemic diads such as i + 1,i+ 2 are represented by Ur. These statistical weight

matrices can be written as:

UV ¼1 1

1 0

24

35 Um ¼

g2xW g

g x

24

35 Ur ¼

g2 gxV

gxV 1

24

35 ð26Þ

with the rotational isomers indexed in the order trans (/c 180j), gauche (/c 300j).Values of the conformational energies required to compute the statistical weights

appearing in Eq. (26) are summarized in Table 2. They were taken from the literature in the

case of PVCz [32,33]. However, although some calculations of these parameters for PVP

have been published [20] for a slightly different conformational model, we have preferred

to compute them by means of molecular dynamics simulations. The DL_POLY package

[34] was employed in all the MD simulations. A time step d = 2 fs (i.e., 2� 10� 15 s) was

used for the integration of the equations of motion. The temperature of the system was

kept constant by means of a Nose–Hoover thermostat [35] with a relaxation time of 500

fs. The AMBER force field [36] was used in the present work. Coulombic interactions

were computed as pairwise interactions among partial charges assigned to every atom of

the system by means of the MOPAC-AM1 procedure [37] employing a distance-dependent

dielectrical constant.

Two oligomers, each one of them containing eight repeat units of PVP, were studied.

One of the oligomers was meso and the second one racemic. Their structure were first

Table 2

Conformational parameters required to compute the statistical weight matrices (Eq. (26)) for PVCz (taken from

Refs. [20,33]) and PVP (calculated from MD simulation) chains

Statistical Conformational Meaning Value

weight energyPVCz PVP

g Eg first-order t vs. g � 0.85 0.15

x Ex second-order CH2. . .CH2 0.5 lxV ExV second-order CH2. . .R 0.5 1.29

xW ExW second-order R. . .R 2.0 l

Energies in kcal/mol.


optimized by minimizing the energy as a function of all internal coordinates, i.e., bond

lengths, bond angles and torsions. The MD simulation was then started. The first part of

the simulation was a thermostating process in which the temperature of the system was

raised from 0 to 300 K with increments of 50 K and allowing 500 fs for equilibration at

each new temperature. After the working temperature was reached, the simulation was

continued for 4 ns during which the rotational states of the three CHR–CH2–CHR pairs of

bonds at the center of the oligomer were recorded with intervals of 100 fs, thus producing

1.2� 105 conformations for this kind of bond pairs. Two pairs of bonds were disregarded

at each extreme of the oligomer to avoid end effects. The analysis of the recorded

conformations consisted of counting how many times along the MD trajectory the pair of

bonds adopted a given value. That number divided by the total number of analyzed

conformations provides the probability of occurrence for that conformation. The results are

shown in Fig. 9.

Inspection of the probability surface for the meso diad (top panel on Fig. 9) reveals that

only the symmetrically equivalent tg and gt conformations are allowed for this pair of

bonds which indicates that second order weights x and xW appearing, respectively, in gg

and tt conformations are very small, i.e., their corresponding energies very high. Under

these circumstances and as far as the meso diad is concerned, the value of g is irrelevant

since both allowed states have the same statistical weight according to Eq. (26). In the case

of the racemic diad (lower panel on Fig. 9), the allowed conformations are tt which

represents a fraction of ca. 0.34 of the total conformations, gg with a fraction of about

0.56, and the equivalent tg and gt, each one of them with a fraction of ca. 0.05. These

results, together with the expression of Ur given in Eq. (26) indicates that g2c 0.34/0.56

and gxVc 0.05/0.56 and, since the calculations were performed at 300 K, Egc 0.15,

ExV = 1.29, both in kcal/mol. The values of these conformational energies are summarized

in the last column of Table 2.

It is interesting to compare the numerical values of the statistical weight matrices for

PVCz and PVP once their corresponding values of conformational energies have been

introduced. After renormalization, these matrices are:

UmðPVCzÞ ¼0:063 0:445

0:445 0:047

24

35 UrðPVCzÞ ¼

0:789 0:082

0:082 0:047

24

35

UmðPVPÞ ¼0:000 0:500

0:500 0:000

24

35 UrðPVPÞ ¼

0:338 0:050

0:050 0:562

24

35

ð27Þ

The conformational characteristics imposed by these matrices are similar (after all, both

PVCz and PVP are vinyl polymers with bulky and roughly planar side groups). However,

there are some significant differences arising from the fact that the carbazole group is

planar while the pyrrolidone group contains H atoms going out the plane of the ring. Thus,

two consecutive carbazole groups could minimize their interaction by aligning parallel to

each other, while in the case of pyrrolidone group, this minimization is not possible, and

consequently the conformations tt and gg are not allowed in the meso diad of the PVP

chain. At any rate, the helical tg conformation is the preferred one (in the case of PVCz) or

Fig. 9. Probability distribution for the pair of bonds CHR–CH2–CHR within meso and racemic diads of the PVP

chain obtained from MD simulations performed at 300 K. See text for details.


the only one allowed (in the case of PVP) for meso sequences. This sequence is disrupted

by the racemic units which most of the times are in the extended tt conformation in the

case of PVCz, while in PVP, the disruption is produced by the most compact gg

conformation, and only one out of each three times will the tt appear. Consequently, the

PVP chain should be more compact than its PVCz counterpart.

Calculations of the averaged molecular dimensions were performed employing Monte

Carlo sampling techniques. Thus, chains of either PVCz or PVP containing a predeter-

mined number of repeat units X (i.e., number of skeletal bonds n = 2X), and a fixed fraction


of meso units wm with random placement of meso and racemic centers was generated.

Then, Eqs. (25) and (27) were employed to compute the a priori probabilities for all the

allowed orientations of each pair of bonds within the generated chain. Next, 1000 different

conformations of the chain were produced according to those a priori probabilities and

their molecular dimensions evaluated by serial products of generator matrices. The

generation of the chains was repeated 1000 times thus providing a total number of 106

conformations for each polymer. Values of the averages over all those conformations are

given below. The standard error on those averages amounted to ca. 1%. Calculations were

performed at 300 K for chains with number of repeat units X ranging from 10 to 300 and

wm = 0.2 for PVCz and 0.5 for PVP which are the approximate content of meso units in

samples of those polymers obtained by free radical polymerization [20,32,33].

Values of the squared end-to-end vector hr2io computed as explained above were

converted into the characteristic ratio Cn and are shown in Fig. 10 as a function of the

number of repeat units X. As this figure indicates, Cn increases with X for short chains and

tends to asymptotic limits of ca. 16.5 and 11.0, respectively, for PVCz and PVP at

moderately long chains with ca. Xc 200. The value obtained for PVCz is in very good

agreement with the experimental result. However, in the case of PVP, the theoretical result

is smaller than the experimental one.

The averages of the end-to-end vector hri and its components in a coordinate system

affixed to the first bond of the chain are represented as function of X in Fig. 11. By

definition, the hxi component is the persistence length a. However, when applied to freely

Fig. 10. Values of the characteristic ratio of the unperturbed dimensions as a function of the number of repeat units

calculated at 300 K with the RIS model for PVCz and PVP chains.

Fig. 11. Averages of the end-to-end vector hri and its components in the coordinate system affixed to the first

skeletal bond as a function of the number for repeat units calculated at 300 K with the RIS model for PVCz (solid

lines) and PVP (dot lines). hri: (E); hxi: (.); hyi: (x); hzi: (+).


rotating chains, hyi and hzi components vanish and the value of a is independent of the

orientation of the coordinate system. This is not the case in more rigid chains as Fig. 11

indicates. Consequently, the persistence length obtained by identifying a with hxi woulddepend on the orientation of the first bond. For this reason, it is more convenient to employ

the value of the modulus hri that is usually referred to as the persistence vector, which doesnot depend on the coordinate system. It can be observed in Fig. 6 that hri and its

components reach constant values for rather short chains, i.e., Xc 50. The values of hri,hxi and hyi components are roughly twice in the case of PVCz than their counterparts in

the PVP chain. However, the hzi component is larger, in absolute value, for PVP. Thus, the

box defined by the three components is much closer to a cube in the case of PVP than in

PVCz, and therefore, the average shape of this polymer is much closer to globular than in

the case of PVCz.

5. Conclusions

The SEC–MALS technique affords a complete analysis of the solution properties of a

polymer solution, including the extrapolation of molecular dimensions to unperturbed

conditions, employing just a single polydisperse sample. The procedure has been applied

to study solutions of PVCz in THF and PVP in water with 0.1 M concentration of NaNO3,


both at 25 jC, as examples of systems which are, respectively, above and below

unperturbed conditions. Values of the unperturbed dimensions were also obtained with

a realistic RIS model which allows a better understanding of the behavior of these two

polymeric chains. Theoretical results are in very good agreement with the experimental

values in the case of PVCz, although the agreement is poorer in the case of PVP.

Acknowledgements

Financial support from projects BQU2001-1158 and UAH2002/34 are gratefully

acknowledged.

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