# Density Functional Theory of Polymer Structure and Conformations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Approach

## 3. Results and Discussion

^{4}in θ solvent. It seems that a good agreement can be achieved between DFT calculations and the experimental values [48], whereas the Debye expression is not always valid in the fractal regime ($q\ge 1.0$). This can be seen by the gradual transition from a slope of −2 for the Debye equation to a slope of −1 for PS chains. Figure 4b shows the form factor of several different lengths of PS melt. In particular, we are interested in the so-called fractal regime. This regime provides information related to the chain statistics inside the coil, which can reveal details about stiffness and self-avoiding behavior. In the regime, the results given by two equations are closer as the polymerization degree increases. One can expect that on a scale which is large enough, they will again appear as flexible coils.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the polymer model used in this work. Here, a middle segment from a polymer chain (filled black sphere) is fixed at the origin. The density distributions of segments from the tethered fragments ($C$ and $D$), free ($F$), and short ($S$) chain molecules are related to the intra- and intermolecular segment–segment correlation functions. In a homogeneous polymer melt, the short chain no longer exists.

**Figure 2.**Comparison of the average (

**a**) inter- and (

**b**) intramolecular correlation functions of the PS chain obtained from theory and molecular simulations [42] at 413.2 K.

**Figure 3.**(

**a**) Gyration radius as a function of chain length for PS chains at 500 K; (

**b**) Determination of the scaling exponents. The circles are calculated results and the solid line is the best linear regression of the circles.

**Figure 4.**(

**a**) Comparison between experimental data of the form factors and calculated ones for PS melt with the molecular weight 1 × 10

^{4}in θ solvent; (

**b**) Form factor for different lengths of PS melt at 500 K. Solid lines correspond to DFT calculations in Equation (21). Symbols show the curve obtained from the Debye expression.

**Figure 6.**(

**a**) Gyration radius as a function of chain length for PEO and PMMA melts at 300 K; (

**b**) Determination of the scaling exponents for the two polymers. The squares and circles are calculated results and the solid line is the best linear regression.

**Figure 7.**Gyration radius of PS chain as a function of packing fraction at 500 K. The chain length is fixed at $m=100$.

**Figure 8.**Gyration radius of long chain PS ($m=100$) as a function of chain length of short chain PS (${m}_{1}=1-100$) at 500 K.

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**MDPI and ACS Style**

Wei, Z.; Ning, N.; Zhang, L.; Tian, M.; Mi, J.
Density Functional Theory of Polymer Structure and Conformations. *Polymers* **2016**, *8*, 121.
https://doi.org/10.3390/polym8040121

**AMA Style**

Wei Z, Ning N, Zhang L, Tian M, Mi J.
Density Functional Theory of Polymer Structure and Conformations. *Polymers*. 2016; 8(4):121.
https://doi.org/10.3390/polym8040121

**Chicago/Turabian Style**

Wei, Zhaoyang, Nanying Ning, Liqun Zhang, Ming Tian, and Jianguo Mi.
2016. "Density Functional Theory of Polymer Structure and Conformations" *Polymers* 8, no. 4: 121.
https://doi.org/10.3390/polym8040121